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Physical Review B

Extinction of the Berezinskii-Kosterlitz-Thouless phase transition by nonmagnetic disorder in planar symmetry spin models

Gary Mathew Wysin, A. R. Pereira, Ivo A. Marques, S. A. Leonel, P. Z. Coura

Published Sep 15, 2005• 50 citations
Publisher

Abstract

The Berezinskii-Kosterlitz-Thouless (BKT) phase 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 ${T}_{c}$ 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 ${T}_{c}$ are consistent with those from the much less precise fourth-order cumulant of Binder. ${T}_{c}$ 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.

DOI
10.1103/physrevb.72.094418

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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 \ensuremathρvac{\ensuremath{\rho}}_{\mathrm{vac}}\ensuremathρvac​ 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 \ensuremathρvac{\ensuremath{\rho}}_{\mathrm{vac}}\ensuremathρvac​, and falls to zero close to the square lattice percolation limit, \ensuremathρvac\ensuremath≈0.41{\ensuremath{\rho}}_{\mathrm{vac}}\ensuremath{\approx}0.41\ensuremathρvac​\ensuremath≈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.
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.
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.
Abstract In a two-dimensional (2D) spin system, the XY model, characterized by planar rotational symmetry, exhibits a unique phenomenon known as the Berezinskii-Kosterlitz-Thouless (BKT) transition. In contrast, the clock model, which introduces discrete rotational symmetry, exhibits the BKT transition at two different temperatures due to this discreteness. In this study, we numerically investigate the BKT transition for XY and six-state clock models over various two-dimensional lattices. We employ two primary methods: the Monte Carlo method, which analyzes the size dependence of the ratio of the correlation functions for two different distances, and a machine-learning approach to classify the different phases --- namely, the low-temperature ordered phase, the intermediate BKT phase, and the high-temperature disordered phase. We identify the BKT transition temperatures for the XY and six-state models on honeycomb, kagome, and diced lattices. Combined with the previously calculated data for the triangular lattice, we then compare these values with the second-order phase transition temperatures of the 2D Ising model, for which exact solutions are known. Our results indicate that the ratio of the BKT transition temperatures for each lattice relative to the Ising model transition temperatures are close, although the values are not universal.
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.
Abstract In a two-dimensional (2D) spin system, the XY model, characterized by planar rotational symmetry, exhibits a unique phenomenon known as the Berezinskii–Kosterlitz–Thouless (BKT) transition. In contrast, the clock model, which introduces discrete rotational symmetry, exhibits the BKT transition at two different temperatures due to this discreteness. In this study, we numerically investigate the BKT transition for XY and six-state clock models over various 2D lattices. We employ two primary methods: the Monte Carlo method, which analyzes the size dependence of the ratio of the correlation functions for two different distances, and a machine-learning approach to classify the different phases—namely, the low-temperature ordered phase, the intermediate BKT phase, and the high-temperature disordered phase. We identify the BKT transition temperatures for the XY and six-state models on honeycomb, kagome, and diced lattices. Combined with the previously calculated data for the triangular lattice, we then compare these values with the second-order phase transition temperatures of the 2D Ising model, for which exact solutions are known. Our results indicate that the ratio of the BKT transition temperatures for each lattice relative to the Ising model transition temperatures are close, although the values are not universal.
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.