Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0503562 · Physical Review B

Kosterlitz-Thouless transition in planar spin models with bond dilution

Tasrief Surungan, Yutaka Okabe

Published May 31, 2005• 30 citations
arXivHTMLPDF

Abstract

We study the two-dimensional bond-diluted $XY$ and six-state clock models by Monte Carlo simulation with cluster spin updates. Various concentrations of depleted bonds were simulated, in which we found a systematic decrease of the Kosterlitz-Thouless (KT) transition temperatures of both $XY$ and six-state clock models as the concentration of dilution decreases. For the six-state clock model, a second KT transition at lower temperature was observed. The KT transition temperatures as well as the decay exponent $\ensuremath{\eta}$ for each concentration of dilution are estimated. It is observed that the quasi-long-range order disappears at the concentration of dilution very close to the percolation threshold. The decay exponent $\ensuremath{\eta}$ of the KT transitions calculated at each concentration indicates that the universality class belongs to the pure $XY$ and clock models, analogous to the expectation of the Harris criterion for the irrelevance of randomness in the continuous phase transition of systems with nondiverging specific heat.

DOI
10.1103/physrevb.71.184438

Related papers

Phase Transition of Two-Dimensional Diluted XY and Clock Models

Yutaka Okabe, Tasrief Surungan

We study the two-dimensional bond-diluted XY and 6-state clock models by Monte Carlo simulation with cluster spin updates. We show that with dilution the KT transition temperature approaches zero at the percolation threshold, and the decay exponent at the KT temperature is universal, that is, &eegr;= 1/4. For the 6-state clock model, we observe a systematic shift of lower critical temperature due to the discrete symmetry. The universal behavior of the decay exponent at the lower KT temperature is also shown.

Progress of Theoretical Physics Supplement•Jan 1, 2005•2 citations

Critical behavior of two-dimensional spin systems under the random-bond six-state clock model

Raymond P. H. Wu, Veng Cheong Lo, Haitao Huang

The critical behavior of the clock model in two-dimensional square lattice is studied numerically using Monte Carlo method with Wolff algorithm. The Kosterlitz-Thouless (KT) transition is observed in the six-state clock model, where an intermediate phase exists between the low-temperature ordered phase and the high-temperature disordered phase. The bond randomness is introduced to the system by assuming a Gaussian distribution for the coupling coefficients with the mean μ=1 and different values of variance, from σ2=0.1 to σ2=3.0. An abrupt jump in the helicity modulus at the transition, which is the key characteristic of the KT transition, is verified with a stability argument. The critical temperature Tc for both pure and disordered systems is determined from the critical exponent η(Tc)=1/4. The results showed that a small amount of disorder (small σ) reduces the critical temperature of the system, without altering the nature of transition. However, a larger amount of disorder changes the transition from the KT-type into that of non-KT-type.

Journal of Applied Physics•Sep 15, 2012•8 citations

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 +2 more

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.

Physical Review B•Sep 15, 2005•50 citations

Kosterlitz-Thouless Transition of the XY model on rewired square lattices

Tasrief Surungan

The ferromagnetic XY model (O(2) symmetry) on decorated square lattices is \nstudied via Monte Carlo simulation to probe the role played by an increase in coordination \nnumber on the existing Kosterlitz-Thouless (KT) transition [1]. It is a unique transition between \nthe quasi-long range order (QLRO), also called the KT phase, and the paramagnetic phase at \nhigher temperature [2, 3]. The QLRO is the only type of order allowed by theMermin-Hohenberg \ntheorem for two dimensional (2D) spin model with continuous symmetry [4]. In diluted square \nlattices, where some links are depleted, it was found that the KT phase remains so long as the \nlinks are percolated. Here we probe whether the increase in average number of links preserves \nor destroys the KT phase. The decorated lattice is obtained by adding an extra link to each site \nof a square lattice, randomly connecting each site to one of its next-nearest neighbors [5]. The \naverage number of nearest links increases from two to three, which corresponds to an increase \nin dimensionality. The existence of KT phase is characterized by the correlation ratio, which \nwas also used in characterizing the critical properties of the 2D edge-cubic model [6]. It was \nfound that the KT phase shifted to higher temperature, which indicates that the KT phase on \ndecorated lattices is more robust against thermal fluctuation compared to that of the native \nsquare lattice. [3]. \nReferences

•Aug 14, 2016•0 citations

Bond dilution in the 3D Ising model: a Monte Carlo study

Pierre-Emmanuel Berche, Christophe Chatelain, Wolfhard Janke

We study by Monte Carlo simulations the influence of bond dilution on the three-dimensional Ising model. This paradigmatic model in its pure version displays a second-order phase transition with a positive specific heat critical exponent $\alpha$ . According to the Harris criterion disorder should hence lead to a new fixed point characterized by new critical exponents. We have determined the phase diagram of the diluted model, starting from the pure model limit down to the neighbourhood of the percolation threshold. For the estimation of critical exponents, we have first performed a finite-size scaling study, where we concentrated on three different dilutions to check the stability of the disorder fixed point. We emphasize in this work the great influence of the cross-over phenomena between the pure, disorder and percolation fixed points which lead to effective critical exponents dependent on the concentration. In a second set of simulations, the temperature behaviour of physical quantities has been studied in order to characterize the disorder fixed point more accurately. In particular this allowed us to estimate ratios of some critical amplitudes. In accord with previous observations for other models this provides stronger evidence for the existence of the disorder fixed point since the amplitude ratios are more sensitive to the universality class than the critical exponents. Moreover, the question of non-self-averaging at the disorder fixed point is investigated and compared with recent results for the bond-diluted q = 4 Potts model. Overall our numerical results provide evidence that, as expected on theoretical grounds, the critical behaviour of the bond-diluted model is indeed governed by the same universality class as the site-diluted model.

•Jan 1, 2008•44 citations

Phase diagram of six-state clock model on rewired square lattices

Tasrief Surungan

Abstract The six-state clock model (SSCM) on rewired square lattice is studied using Monte Carlo simulation with Wang-Landau algorithm. This is a discrete counterpart of the well-known XY model, the native host of a unique topological phase transition called Kosterlitz-Tholess (KT) transition. The model has two KT transitions, i.e., at temperature T 1 and T 2, where T 1 < T 2. The first transition separates the lower temperature magnetic order and the quasi-long range order (QLRO) also known as KT phase; while the second transition separates the QLRO and the higher temperature paramagnetic phase. It has been established that the presence of KT phase is affected by the presence of randomness in the form of site and bond dilution. This intermediate phase is totally ruled out if bonds or sites of the lattice are no longer percolated. Here different type of randomness is probed, namely the rewired lattices, obtained by randomly adding one extra bond to each lattice site, and connect the site to one of its next-nearest neighbors. As a results, the average number of neighbors C increases. The increase of C affects the existence of KT phase. For each value of C, the KT temperatures, T 1 and T 2, were estimated from the plot of specific heats. Variation of KT temperatures for different values of C is observed, which is plotted with respect to each corresponding C to obtain the system phase diagram.

Journal of Physics Conference Series•Oct 1, 2019•1 citation
We study the two-dimensional bond-diluted XYXYXY and six-state clock models by Monte Carlo simulation with cluster spin updates. Various concentrations of depleted bonds were simulated, in which we found a systematic decrease of the Kosterlitz-Thouless (KT) transition temperatures of both XYXYXY and six-state clock models as the concentration of dilution decreases. For the six-state clock model, a second KT transition at lower temperature was observed. The KT transition temperatures as well as the decay exponent \ensuremathη\ensuremath{\eta}\ensuremathη for each concentration of dilution are estimated. It is observed that the quasi-long-range order disappears at the concentration of dilution very close to the percolation threshold. The decay exponent \ensuremathη\ensuremath{\eta}\ensuremathη of the KT transitions calculated at each concentration indicates that the universality class belongs to the pure XYXYXY and clock models, analogous to the expectation of the Harris criterion for the irrelevance of randomness in the continuous phase transition of systems with nondiverging specific heat.
We study the two-dimensional bond-diluted XY and 6-state clock models by Monte Carlo simulation with cluster spin updates. We show that with dilution the KT transition temperature approaches zero at the percolation threshold, and the decay exponent at the KT temperature is universal, that is, &eegr;= 1/4. For the 6-state clock model, we observe a systematic shift of lower critical temperature due to the discrete symmetry. The universal behavior of the decay exponent at the lower KT temperature is also shown.
The critical behavior of the clock model in two-dimensional square lattice is studied numerically using Monte Carlo method with Wolff algorithm. The Kosterlitz-Thouless (KT) transition is observed in the six-state clock model, where an intermediate phase exists between the low-temperature ordered phase and the high-temperature disordered phase. The bond randomness is introduced to the system by assuming a Gaussian distribution for the coupling coefficients with the mean μ=1 and different values of variance, from σ2=0.1 to σ2=3.0. An abrupt jump in the helicity modulus at the transition, which is the key characteristic of the KT transition, is verified with a stability argument. The critical temperature Tc for both pure and disordered systems is determined from the critical exponent η(Tc)=1/4. The results showed that a small amount of disorder (small σ) reduces the critical temperature of the system, without altering the nature of transition. However, a larger amount of disorder changes the transition from the KT-type into that of non-KT-type.
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.
The ferromagnetic XY model (O(2) symmetry) on decorated square lattices is \nstudied via Monte Carlo simulation to probe the role played by an increase in coordination \nnumber on the existing Kosterlitz-Thouless (KT) transition [1]. It is a unique transition between \nthe quasi-long range order (QLRO), also called the KT phase, and the paramagnetic phase at \nhigher temperature [2, 3]. The QLRO is the only type of order allowed by theMermin-Hohenberg \ntheorem for two dimensional (2D) spin model with continuous symmetry [4]. In diluted square \nlattices, where some links are depleted, it was found that the KT phase remains so long as the \nlinks are percolated. Here we probe whether the increase in average number of links preserves \nor destroys the KT phase. The decorated lattice is obtained by adding an extra link to each site \nof a square lattice, randomly connecting each site to one of its next-nearest neighbors [5]. The \naverage number of nearest links increases from two to three, which corresponds to an increase \nin dimensionality. The existence of KT phase is characterized by the correlation ratio, which \nwas also used in characterizing the critical properties of the 2D edge-cubic model [6]. It was \nfound that the KT phase shifted to higher temperature, which indicates that the KT phase on \ndecorated lattices is more robust against thermal fluctuation compared to that of the native \nsquare lattice. [3]. \nReferences
We study by Monte Carlo simulations the influence of bond dilution on the three-dimensional Ising model. This paradigmatic model in its pure version displays a second-order phase transition with a positive specific heat critical exponent α\alphaα . According to the Harris criterion disorder should hence lead to a new fixed point characterized by new critical exponents. We have determined the phase diagram of the diluted model, starting from the pure model limit down to the neighbourhood of the percolation threshold. For the estimation of critical exponents, we have first performed a finite-size scaling study, where we concentrated on three different dilutions to check the stability of the disorder fixed point. We emphasize in this work the great influence of the cross-over phenomena between the pure, disorder and percolation fixed points which lead to effective critical exponents dependent on the concentration. In a second set of simulations, the temperature behaviour of physical quantities has been studied in order to characterize the disorder fixed point more accurately. In particular this allowed us to estimate ratios of some critical amplitudes. In accord with previous observations for other models this provides stronger evidence for the existence of the disorder fixed point since the amplitude ratios are more sensitive to the universality class than the critical exponents. Moreover, the question of non-self-averaging at the disorder fixed point is investigated and compared with recent results for the bond-diluted q = 4 Potts model. Overall our numerical results provide evidence that, as expected on theoretical grounds, the critical behaviour of the bond-diluted model is indeed governed by the same universality class as the site-diluted model.
Abstract The six-state clock model (SSCM) on rewired square lattice is studied using Monte Carlo simulation with Wang-Landau algorithm. This is a discrete counterpart of the well-known XY model, the native host of a unique topological phase transition called Kosterlitz-Tholess (KT) transition. The model has two KT transitions, i.e., at temperature T 1 and T 2, where T 1 < T 2. The first transition separates the lower temperature magnetic order and the quasi-long range order (QLRO) also known as KT phase; while the second transition separates the QLRO and the higher temperature paramagnetic phase. It has been established that the presence of KT phase is affected by the presence of randomness in the form of site and bond dilution. This intermediate phase is totally ruled out if bonds or sites of the lattice are no longer percolated. Here different type of randomness is probed, namely the rewired lattices, obtained by randomly adding one extra bond to each lattice site, and connect the site to one of its next-nearest neighbors. As a results, the average number of neighbors C increases. The increase of C affects the existence of KT phase. For each value of C, the KT temperatures, T 1 and T 2, were estimated from the plot of specific heats. Variation of KT temperatures for different values of C is observed, which is plotted with respect to each corresponding C to obtain the system phase diagram.