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 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 PhysicsSep 15, 20128 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.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 SeriesOct 1, 20191 citation

Probability-changing cluster algorithm for two-dimensional XY and clock models

Yusuke Tomita, Yutaka Okabe

We extend the newly proposed probability-changing cluster (PCC) Monte Carlo algorithm to the study of systems with the vector order parameter. Wolff's idea of the embedded cluster formalism is used for assigning clusters. The Kosterlitz-Thouless (KT) transitions for the two-dimensional (2D) $\mathrm{XY}$ and q-state clock models are studied by using the PCC algorithm. Combined with the finite-size scaling analysis based on the KT form of the correlation length, $\ensuremath{\xi}\ensuremath{\propto}\mathrm{exp}(c/\sqrt{{T/T}_{\mathrm{KT}}\ensuremath{-}1}),$ we determine the KT transition temperature and the decay exponent $\ensuremath{\eta}$ as ${T}_{\mathrm{KT}}=0.8933(6)$ and $\ensuremath{\eta}=0.243(4)$ for the 2D $\mathrm{XY}$ model. We investigate two transitions of the KT type for the 2D q-state clock models with $q=6,8,12$ and confirm the prediction of $\ensuremath{\eta}{=4/q}^{2}$ at ${T}_{1},$ the low-temperature critical point between the ordered and $\mathrm{XY}$ -like phases, systematically.We extend the newly proposed probability-changing cluster (PCC) Monte Carlo algorithm to the study of systems with the vector order parameter. Wolff's idea of the embedded cluster formalism is used for assigning clusters. The Kosterlitz-Thouless (KT) transitions for the two-dimensional (2D) $\mathrm{XY}$ and q-state clock models are studied by using the PCC algorithm. Combined with the finite-size scaling analysis based on the KT form of the correlation length, $\ensuremath{\xi}\ensuremath{\propto}\mathrm{exp}(c/\sqrt{{T/T}_{\mathrm{KT}}\ensuremath{-}1}),$ we determine the KT transition temperature and the decay exponent $\ensuremath{\eta}$ as ${T}_{\mathrm{KT}}=0.8933(6)$ and $\ensuremath{\eta}=0.243(4)$ for the 2D $\mathrm{XY}$ model. We investigate two transitions of the KT type for the 2D q-state clock models with $q=6,8,12$ and confirm the prediction of $\ensuremath{\eta}{=4/q}^{2}$ at ${T}_{1},$ the low-temperature critical point between the ordered and $\mathrm{XY}$-like phases, systematically.

arXiv:cond-mat/0202161Physical review. B, Condensed matterApr 12, 200291 citations

Phase transitions in the two-dimensional XY model with random phases: A Monte Carlo study

J. Maucourt, Daniel R. Grempel

We study the two-dimensional $\mathrm{XY}$ 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 $T=0$ 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.We study the two-dimensional $\mathrm{XY}$ 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 $T=0$ 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.

arXiv:cond-mat/9703109Physical review. B, Condensed matterAug 1, 199730 citations

Critical behavior of the six-state clock model in two dimensions

Murty S. S. Challa, David Paul Landau

We have examined the critical properties of the q=6 clock (vector Potts) model in two dimensions through Monte Carlo simulations. The model was investigated on L\ifmmode\times\else\texttimes\fi{}L square lattices of size L=4 to L=72 with periodic boundary conditions. We found an intermediate XY-like phase between a low-temperature ordered phase and a high-temperature disordered phase. The phase transitions occur at ${k}_{B}$ ${T}_{1}$ /J=0.68\ifmmode\pm\else\textpm\fi{}0.02 and ${k}_{B}$ ${T}_{2}$ /J=0.92\ifmmode\pm\else\textpm\fi{}0.01 and are of the Kosterlitz-Thouless type. The susceptibility diverges in the intermediate phase and the exponent \ensuremath{\eta} varies between 0.100 at ${T}_{1}$ and 0.275 at ${T}_{2}$ .We have examined the critical properties of the q=6 clock (vector Potts) model in two dimensions through Monte Carlo simulations. The model was investigated on L\ifmmode\times\else\texttimes\fi{}L square lattices of size L=4 to L=72 with periodic boundary conditions. We found an intermediate XY-like phase between a low-temperature ordered phase and a high-temperature disordered phase. The phase transitions occur at ${k}_{B}$${T}_{1}$/J=0.68\ifmmode\pm\else\textpm\fi{}0.02 and ${k}_{B}$${T}_{2}$/J=0.92\ifmmode\pm\else\textpm\fi{}0.01 and are of the Kosterlitz-Thouless type. The susceptibility diverges in the intermediate phase and the exponent \ensuremath{\eta} varies between 0.100 at ${T}_{1}$ and 0.275 at ${T}_{2}$.

Physical review. B, Condensed matterJan 1, 1986145 citations

Monte Carlo analysis of the phase transitions in the two-dimensional J1−J2XY model

Damien Loison, Pascal Simon

We consider the two-dimensional (2D) ${J}_{1}\ensuremath{-}{J}_{2}$ classical $\mathrm{XY}$ model on a square lattice. In the frustrated phase corresponding to ${J}_{2}>{J}_{1}/2,$ an Ising-like order parameter emerges by an ``order due to disorder'' effect. This leads to a discrete ${Z}_{2}$ symmetry plus the $U(1)$ global one. Using a powerful algorithm we show that the system undergoes two transitions at different but still very close temperatures, one of Kosterlitz-Thouless (KT) type and another one which does not belong to the expected Ising universality class. A new analysis of the KT transition has been developed in order to avoid the use of the nonuniversal helicity jump and to allow the computation of the exponents without a precise determination of the critical temperature. Moreover, our huge number of data enables us to exhibit the existence of large finite size effects explaining the dispersed results found in the literature concerning the more studied frustrated 2D, $\mathrm{XY}$ models.We consider the two-dimensional (2D) ${J}_{1}\ensuremath{-}{J}_{2}$ classical $\mathrm{XY}$ model on a square lattice. In the frustrated phase corresponding to ${J}_{2}>{J}_{1}/2,$ an Ising-like order parameter emerges by an ``order due to disorder'' effect. This leads to a discrete ${Z}_{2}$ symmetry plus the $U(1)$ global one. Using a powerful algorithm we show that the system undergoes two transitions at different but still very close temperatures, one of Kosterlitz-Thouless (KT) type and another one which does not belong to the expected Ising universality class. A new analysis of the KT transition has been developed in order to avoid the use of the nonuniversal helicity jump and to allow the computation of the exponents without a precise determination of the critical temperature. Moreover, our huge number of data enables us to exhibit the existence of large finite size effects explaining the dispersed results found in the literature concerning the more studied frustrated 2D, $\mathrm{XY}$ models.

arXiv:cond-mat/9912438Physical review. B, Condensed matterMar 1, 200041 citations

\mathrm{XY}

and q-state clock models are studied by using the PCC algorithm. Combined with the finite-size scaling analysis based on the KT form of the correlation length, $\ensuremath{\xi}\ensuremath{\propto}\mathrm{exp}(c/\sqrt{{T/T}_{\mathrm{KT}}\ensuremath{-}1}),$ we determine the KT transition temperature and the decay exponent $\ensuremath{\eta}$ as

{T}_{\mathrm{KT}}=0.8933(6)

and $\ensuremath{\eta}=0.243(4)$ for the 2D

\mathrm{XY}

model. We investigate two transitions of the KT type for the 2D q-state clock models with

q=6,8,12

and confirm the prediction of $\ensuremath{\eta}{=4/q}^{2}$ at

{T}_{1},

the low-temperature critical point between the ordered and

\mathrm{XY}

-like phases, systematically.

We study the two-dimensional

\mathrm{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=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.

{k}_{B}

{T}_{1}

/J=0.68\ifmmode\pm\else\textpm\fi{}0.02 and

{k}_{B}

{T}_{2}

/J=0.92\ifmmode\pm\else\textpm\fi{}0.01 and are of the Kosterlitz-Thouless type. The susceptibility diverges in the intermediate phase and the exponent \ensuremath{\eta} varies between 0.100 at

{T}_{1}

and 0.275 at

{T}_{2}

We consider the two-dimensional (2D) ${J}_{1}\ensuremath{-}{J}_{2}$ classical

\mathrm{XY}

model on a square lattice. In the frustrated phase corresponding to

{J}_{2}>{J}_{1}/2,

an Ising-like order parameter emerges by an ``order due to disorder'' effect. This leads to a discrete

{Z}_{2}

symmetry plus the

U(1)

global one. Using a powerful algorithm we show that the system undergoes two transitions at different but still very close temperatures, one of Kosterlitz-Thouless (KT) type and another one which does not belong to the expected Ising universality class. A new analysis of the KT transition has been developed in order to avoid the use of the nonuniversal helicity jump and to allow the computation of the exponents without a precise determination of the critical temperature. Moreover, our huge number of data enables us to exhibit the existence of large finite size effects explaining the dispersed results found in the literature concerning the more studied frustrated 2D,

\mathrm{XY}

models.