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

Journal of Physics Condensed Matter

Monte Carlo simulations of finite-size effects in Kosterlitz Thouless systems

F. Falo, L. M. Florı́a, R Navarro

Published Aug 7, 1989• 6 citations
Publisher

Abstract

The two-dimensional planar rotator and XY ferromagnetic models for finite lattice sizes (triangular network), with free and periodic boundary conditions, have been studied using Monte Carlo simulations. Thermodynamic quantities, suitable for comparison with experimental data, have been obtained. Heat capacity data are almost insensitive to the lattice size and to the type of boundary conditions, whereas susceptibility data are sensitive to both the lattice size and the boundary conditions. For the Kosterlitz-Thouless phase transition non-universal critical parameters are inferred. They depend on the lattice, the model and the boundary conditions. It is conjectured that the main differences between free and periodic boundary conditions are caused by the influence of the free surface on vortex energetics.

DOI
10.1088/0953-8984/1/31/012

Related papers

The Kosterlitz–Thouless transition in thin films: a Monte Carlo study of three-dimensional lattice models

Martin Hasenbusch

We study the phase transition of thin films in the three-dimensional XY universality class. To this end, we perform a Monte Carlo study of the improved two-component ϕ 4 model, the improved dynamically diluted XY model and the standard XY model on the simple cubic lattice. We study films of a thickness up to L 0 = 32 lattice spacings. In the short direction of the lattice, free boundary conditions are employed. Using a finite size scaling (FSS) method, proposed recently, we determine the transition temperature with high accuracy. The effectively two-dimensional finite size scaling behaviour of the Binder cumulant U 4 , the second-moment correlation length over the lattice size ξ 2nd / L , the ratio of the partition functions with anti-periodic and periodic boundary conditions Z a / Z p and the helicity modulus ϒ clearly confirm the Kosterlitz–Thouless nature of the transition. We analyse the scaling of the transition temperature with the thickness L 0 of the film. We compute the universal ratio of the thickness of the film L 0 and the transverse correlation length ξ T in the three-dimensional thermodynamic limit at the Kosterlitz–Thouless transition temperature of a film of thickness L 0 : [ L 0,KT /ξ T ] * = 1.595(7). This results can be compared with experimental results on thin films of 4 He near the λ-transition.

arXiv:0811.2178•Journal of Statistical Mechanics Theory and Experiment•Feb 2, 2009•21 citations

Monte Carlo Simulation of the Antiferromagnetic Ising Model on the Triangular Lattice with the First and Second Neighbour Interactions

Sumiyoshi Fujiki, Keisei Shutoh, Yoshihiko Abe +1 more

Monte Carlo simulations of the Ising model on the triangular lattice with the antiferromagnetic nn interaction J and the ferromagnetic nnn interaction J ' are carried out. Several physical quantities (magnetizations, energies, susceptibilities, specific heats of sublattices and of the total system) are calculated. A phase which resembles the partially disordered phase proposed by Mekata is observed in the finite temperature region. But it is shown in this region that the sublattice magnetizations vanish at the thermodynamic limit and the Kosterlitz-Thouless phase transition is predicted by the finite size analysis.

Journal of the Physical Society of Japan•May 1, 1983•61 citations

Monte Carlo Study of a Triangular Ising Lattice. II. Occurrence of the Kosterlitz-Thouless-Like Phase Transition

Hajime Takayama, Kazuyuki Matsumoto, Hideaki Kawahara +1 more

A two-dimensional triangular Ising lattice with the nearest-neighbor antiferromagnetic interaction J 1 and the next-nearest-neighbor ferromagnetic interaction - J 2 is studied by the Monte Carlo simulation. It is shown that the system with a finite J 2 undergoes the successive phase transition, paramagnetic→Kosterlitz-Thouless (KT)-like→ferrimagnetic phases, as temperature decreases. The sub-lattice switching phenomenon observed in our previous work is shown to be an intrinsic property inherent to the present system which has an XY character. The marginal exponent of the order-parameter correlation function at the KT-like transition point is seen to increase from 1/4 as the ratio J 2 / J 1 decreases, namely, it reveals a deviation from the universal behavior of a genuine XY model.

Journal of the Physical Society of Japan•Aug 1, 1983•65 citations

Numerical simulations of a two-dimensional lattice grain boundary model

A. Jaster, H. H. Hahn

We present detailed Monte Carlo results for a two-dimensional grain boundary model on a lattice. The effective Hamiltonian of the system results from the microscopic interaction of grains with orientations described by spins of unit length, and leads to a nearest-neighbour interaction proportional to the absolute value of the angle between the grains. Our analysis of the correlation length ξ and susceptibility χ in the high-temperature phase favour a Kosterlitz-Thouless-like (KT) singularity over a second-order phase transition. Unconstrained KT fits of χ and ξ confirm the predicted value for the critical exponent ν, while the values of η deviate from the theoretical prediction. Additionally we apply finite-size scaling theory and investigate the question of multiplicative logarithmic corrections to a KT transition. As for the critical exponents our results are similar to data obtained from the XY model, so that both models probably lie in the same universality class. 1

•Jan 1, 1997•3 citations

Finite-temperature states of a two-dimensional frustrated antiferromagnet

Manit Klawtanong, Chatchai Srinitiwarawong

The frustrated antiferromagnetic XY model on the triangular lattice is a simple model describing the critical phenomena of magnetic systems. By using the Monte Carlo simulation method associated with the Metropolis algorithm, we investigate phase transitions of the system at critical temperatures and near critical temperatures. To achieve a scaling form of important parameters, the finite size effect must be taken into account. Our results show clearly that the broken symmetries of the system occur at two separate critical temperatures, the Kosterlitz-Thouless temperature and the Ising-like temperature (TI). However these temperatures lie close to each other about 0.6% compared with the lower temperature. At TI, the reflection symmetry breaking transition occurs and it is known as the Ising-like transition. While at the lower temperature, the rotational symmetry breaking transition occurs and it is known as the Kosterlitz-Thouless transition. Moreover, we found the asymptotic behavior of the correlation length of the Ising-like transition with the critical exponent, ν = 0.74. Taken into account the finite size effect, this value approaches 0.83 which differs from the value, ν = 1, for the Ising universality class. It is likely that the Ising-like transition may not belong to the Ising universality class. Also the other parameters such as the energy of the system, the order parameters, the specific heat, the magnetic susceptibility and the correlation functions have been calculated and their properties near the critical temperatures and at the critical temperatures have been studied.

•Jan 1, 2008•0 citations

Two-dimensionalXXZmodel on a square lattice: A Monte Carlo simulation

Alessandro Cuccoli, V. Tognetti, Ruggero Vaia

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.

Physical review. B, Condensed matter•Oct 1, 1995•88 citations
The two-dimensional planar rotator and XY ferromagnetic models for finite lattice sizes (triangular network), with free and periodic boundary conditions, have been studied using Monte Carlo simulations. Thermodynamic quantities, suitable for comparison with experimental data, have been obtained. Heat capacity data are almost insensitive to the lattice size and to the type of boundary conditions, whereas susceptibility data are sensitive to both the lattice size and the boundary conditions. For the Kosterlitz-Thouless phase transition non-universal critical parameters are inferred. They depend on the lattice, the model and the boundary conditions. It is conjectured that the main differences between free and periodic boundary conditions are caused by the influence of the free surface on vortex energetics.
We study the phase transition of thin films in the three-dimensional XY universality class. To this end, we perform a Monte Carlo study of the improved two-component ϕ 4 model, the improved dynamically diluted XY model and the standard XY model on the simple cubic lattice. We study films of a thickness up to L 0 = 32 lattice spacings. In the short direction of the lattice, free boundary conditions are employed. Using a finite size scaling (FSS) method, proposed recently, we determine the transition temperature with high accuracy. The effectively two-dimensional finite size scaling behaviour of the Binder cumulant U 4 , the second-moment correlation length over the lattice size ξ 2nd / L , the ratio of the partition functions with anti-periodic and periodic boundary conditions Z a / Z p and the helicity modulus ϒ clearly confirm the Kosterlitz–Thouless nature of the transition. We analyse the scaling of the transition temperature with the thickness L 0 of the film. We compute the universal ratio of the thickness of the film L 0 and the transverse correlation length ξ T in the three-dimensional thermodynamic limit at the Kosterlitz–Thouless transition temperature of a film of thickness L 0 : [ L 0,KT /ξ T ] * = 1.595(7). This results can be compared with experimental results on thin films of 4 He near the λ-transition.
Monte Carlo simulations of the Ising model on the triangular lattice with the antiferromagnetic nn interaction J and the ferromagnetic nnn interaction J ' are carried out. Several physical quantities (magnetizations, energies, susceptibilities, specific heats of sublattices and of the total system) are calculated. A phase which resembles the partially disordered phase proposed by Mekata is observed in the finite temperature region. But it is shown in this region that the sublattice magnetizations vanish at the thermodynamic limit and the Kosterlitz-Thouless phase transition is predicted by the finite size analysis.
A two-dimensional triangular Ising lattice with the nearest-neighbor antiferromagnetic interaction J 1 and the next-nearest-neighbor ferromagnetic interaction - J 2 is studied by the Monte Carlo simulation. It is shown that the system with a finite J 2 undergoes the successive phase transition, paramagnetic→Kosterlitz-Thouless (KT)-like→ferrimagnetic phases, as temperature decreases. The sub-lattice switching phenomenon observed in our previous work is shown to be an intrinsic property inherent to the present system which has an XY character. The marginal exponent of the order-parameter correlation function at the KT-like transition point is seen to increase from 1/4 as the ratio J 2 / J 1 decreases, namely, it reveals a deviation from the universal behavior of a genuine XY model.
We present detailed Monte Carlo results for a two-dimensional grain boundary model on a lattice. The effective Hamiltonian of the system results from the microscopic interaction of grains with orientations described by spins of unit length, and leads to a nearest-neighbour interaction proportional to the absolute value of the angle between the grains. Our analysis of the correlation length ξ and susceptibility χ in the high-temperature phase favour a Kosterlitz-Thouless-like (KT) singularity over a second-order phase transition. Unconstrained KT fits of χ and ξ confirm the predicted value for the critical exponent ν, while the values of η deviate from the theoretical prediction. Additionally we apply finite-size scaling theory and investigate the question of multiplicative logarithmic corrections to a KT transition. As for the critical exponents our results are similar to data obtained from the XY model, so that both models probably lie in the same universality class. 1
The frustrated antiferromagnetic XY model on the triangular lattice is a simple model describing the critical phenomena of magnetic systems. By using the Monte Carlo simulation method associated with the Metropolis algorithm, we investigate phase transitions of the system at critical temperatures and near critical temperatures. To achieve a scaling form of important parameters, the finite size effect must be taken into account. Our results show clearly that the broken symmetries of the system occur at two separate critical temperatures, the Kosterlitz-Thouless temperature and the Ising-like temperature (TI). However these temperatures lie close to each other about 0.6% compared with the lower temperature. At TI, the reflection symmetry breaking transition occurs and it is known as the Ising-like transition. While at the lower temperature, the rotational symmetry breaking transition occurs and it is known as the Kosterlitz-Thouless transition. Moreover, we found the asymptotic behavior of the correlation length of the Ising-like transition with the critical exponent, ν = 0.74. Taken into account the finite size effect, this value approaches 0.83 which differs from the value, ν = 1, for the Ising universality class. It is likely that the Ising-like transition may not belong to the Ising universality class. Also the other parameters such as the energy of the system, the order parameters, the specific heat, the magnetic susceptibility and the correlation functions have been calculated and their properties near the critical temperatures and at the critical temperatures have been studied.
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.