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/0501222 · Journal of Physics A Mathematical and General

The square-latticeFmodel revisited: a loop-cluster update scaling study

M Weigel, W Janke

Published Jul 27, 2005• 13 citations
arXivHTMLPDF

Abstract

The six-vertex F model on the square lattice constitutes the unique example of an exactly solved model exhibiting an infinite-order phase transition of the Kosterlitz-Thouless type. As one of the few non-trivial exactly solved models, it provides a welcome gauge for new numerical simulation methods and scaling techniques. In view of the notorious problems of clearly resolving the Kosterlitz-Thouless scenario in the two-dimensional XY model numerically, the F model in particular constitutes an instructive reference case for the simulational description of this type of phase transition. We present a loop-cluster update Monte Carlo study of the square-lattice F model, with a focus on the properties not exactly known such as the polarizability or the scaling dimensions in the critical phase. For the analysis of the simulation data, finite-size scaling is explicitly derived from the exact solution and plausible assumptions. Guided by the available exact results, the careful inclusion of correction terms in the scaling formulae allows for a reliable determination of the asymptotic behaviour.

DOI
10.1088/0305-4470/38/32/002

Related papers

The Binder Cumulant at the Kosterlitz-Thouless Transition, 2008

Martin Hasenbusch B

We study the behaviour of the Binder cumulant on finite square lattices at the Kosterlitz-Thouless phase transition. We determine the fixed point value of the Binder cumulant and the coefficient of the leading logarithmic correction. These calculations are supplemented with Monte Carlo simulations of the classical XY (plane rotator) model, the Villain model and the dual of the absolute value solid-on-solid model. Using the single cluster algorithm, we simulate lattices up to L = 4096. For the lattice sizes reached, subleading corrections are needed to fit the data for the Binder cumulant. We demonstrate that the combined analysis of the Binder cumulant and the second moment correlation length over the lattice size allows for an accurate determination of the Kosterlitz-Thouless transition temperature on relatively small lattices. We test the new method at the example of the 2-component φ 4 model on the lattice.

•Jan 1, 1996•22 citations

Berezinskii-Kosterlitz-Thouless transition in the XY model on the honeycomb lattice: a comprehensive Monte Carlo analysis

Felipe E. F. de Andrade, L. N. Jorge, Cláudio José DaSilva

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.

arXiv:2406.12076•Physica Scripta•May 8, 2025•3 citations

The Binder cumulant at the Kosterlitz–Thouless transition

Martin Hasenbusch

We study the behaviour of the Binder cumulant on finite square lattices at the Kosterlitz–Thouless phase transition. We determine the fixed-point value of the Binder cumulant and the coefficient of the leading logarithmic correction. These calculations are supplemented with Monte Carlo simulations of the classical XY (plane rotator) model, the Villain model and the dual of the absolute value solid-on-solid model. Using the single-cluster algorithm, we simulate lattices up to L = 4096. For the lattice sizes reached, subleading corrections are needed to fit the data for the Binder cumulant. We demonstrate that the combined analysis of the Binder cumulant and the second moment correlation length over the lattice size allows for an accurate determination of the Kosterlitz–Thouless transition temperature on relatively small lattices. We test the new method on the example of the two-component ϕ 4 model on the lattice.

arXiv:0804.1880•Journal of Statistical Mechanics Theory and Experiment•Aug 11, 2008•41 citations

Critical Behaviour of the Six Vertex F-model

N.B. Pannevis

The F-model is one of the six vertex models, a range of two-dimensional models designed to describe the positions of hydrogen atoms in ice. It exhibits a Kosterlitz-Thouless phase transition, and can easily be mapped onto many other models including the roughening transition in the body centered solid-on-solid model. In this thesis the critical behaviour in the F-model is examined through a computational approach, combined with finite-size scaling and real space renormalization. The simulations were run using two Monte Carlo algorithms; the so-called short loop algorithm, and the full lattice cluster algorithm. The ergodicity of the latter is revealed to be problematic, but this is overcome by combining both algorithms. Also, finite-size scaling is applied, returning a critical inverse temperature of \\epsilon \\beta = 0.79 plus or minus 0.02 compared to the known analytical value of log[2]. Furthermore, although the ice rules prevent local renormalization such as by Kadanoff block spins, a functioning coarse graining prescription based on the height function is proposed. This turns out to renormalise the high temperature states into states with temperatures below the critical point, thus preventing the determination of critical exponents from the renormalization flow.

•Aug 29, 2012•1 citation

High-precision Monte Carlo study of the two-dimensionalXYVillain model

Wolfhard Janke E, Klaus Nather

We report two sets of high-precision Monte Carlo simulations of the two-dimensional XY model in Villain's formulation on large square lattices, employing the single-cluster update algorithm. In one set of simulations we use improved estimators to study the correlation length \ensuremath{\xi} and susceptibility \ensuremath{\chi} in the high-temperature phase. On a 1200\ifmmode\times\else\texttimes\fi{}1200 lattice this allows measurements up to \ensuremath{\xi}\ensuremath{\approxeq}140 with statistical errors less than 0.25%. We judge quantitatively the advantage of using improved estimators, estimate autocorrelation times, and compare the numerical efficiency of various definitions of the correlation length. From least-square fits to these data we find clear support for the exponential divergence predicted by Kosterlitz and Thouless. The other set of simulations is performed in the vicinity of the transition point. Here we apply finite-size-scaling theory to obtain an estimate for the exponent \ensuremath{\eta} at criticality.

Physical review. B, Condensed matter•Sep 1, 1993•60 citations

Finite-size scaling in two-dimensional superfluids

Norbert Schultka, Efstratios Manousakis

Using the x-y model and a nonlocal updating scheme called cluster Monte Carlo, we calculate the superfluid density of a two-dimensional superfluid on large-size square lattices L\ifmmode\times\else\texttimes\fi{}L up to 400\ifmmode\times\else\texttimes\fi{}400. This technique allows us to approach temperatures close to the critical point, and by studying a wide range of L values and applying finite-size scaling theory we are able to extract the critical properties of the system. We calculate the superfluid density and from that we extract the renormalization-group beta function. We derive finite-size scaling expressions using the Kosterlitz-Thouless-Nelson renormalization group equations and show that they are in very good agreement with our numerical results. This allows us to extrapolate our results to the infinite-size limit. We also find that the universal discontinuity of the superfluid density at the critical temperature is in very good agreement with the Kosterlitz-Thouless-Nelson calculation and experiments.

arXiv:cond-mat/9310034•Physical review. B, Condensed matter•May 1, 1994•55 citations
The six-vertex F model on the square lattice constitutes the unique example of an exactly solved model exhibiting an infinite-order phase transition of the Kosterlitz-Thouless type. As one of the few non-trivial exactly solved models, it provides a welcome gauge for new numerical simulation methods and scaling techniques. In view of the notorious problems of clearly resolving the Kosterlitz-Thouless scenario in the two-dimensional XY model numerically, the F model in particular constitutes an instructive reference case for the simulational description of this type of phase transition. We present a loop-cluster update Monte Carlo study of the square-lattice F model, with a focus on the properties not exactly known such as the polarizability or the scaling dimensions in the critical phase. For the analysis of the simulation data, finite-size scaling is explicitly derived from the exact solution and plausible assumptions. Guided by the available exact results, the careful inclusion of correction terms in the scaling formulae allows for a reliable determination of the asymptotic behaviour.
We study the behaviour of the Binder cumulant on finite square lattices at the Kosterlitz-Thouless phase transition. We determine the fixed point value of the Binder cumulant and the coefficient of the leading logarithmic correction. These calculations are supplemented with Monte Carlo simulations of the classical XY (plane rotator) model, the Villain model and the dual of the absolute value solid-on-solid model. Using the single cluster algorithm, we simulate lattices up to L = 4096. For the lattice sizes reached, subleading corrections are needed to fit the data for the Binder cumulant. We demonstrate that the combined analysis of the Binder cumulant and the second moment correlation length over the lattice size allows for an accurate determination of the Kosterlitz-Thouless transition temperature on relatively small lattices. We test the new method at the example of the 2-component φ 4 model on 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.
We study the behaviour of the Binder cumulant on finite square lattices at the Kosterlitz–Thouless phase transition. We determine the fixed-point value of the Binder cumulant and the coefficient of the leading logarithmic correction. These calculations are supplemented with Monte Carlo simulations of the classical XY (plane rotator) model, the Villain model and the dual of the absolute value solid-on-solid model. Using the single-cluster algorithm, we simulate lattices up to L = 4096. For the lattice sizes reached, subleading corrections are needed to fit the data for the Binder cumulant. We demonstrate that the combined analysis of the Binder cumulant and the second moment correlation length over the lattice size allows for an accurate determination of the Kosterlitz–Thouless transition temperature on relatively small lattices. We test the new method on the example of the two-component ϕ 4 model on the lattice.
The F-model is one of the six vertex models, a range of two-dimensional models designed to describe the positions of hydrogen atoms in ice. It exhibits a Kosterlitz-Thouless phase transition, and can easily be mapped onto many other models including the roughening transition in the body centered solid-on-solid model. In this thesis the critical behaviour in the F-model is examined through a computational approach, combined with finite-size scaling and real space renormalization. The simulations were run using two Monte Carlo algorithms; the so-called short loop algorithm, and the full lattice cluster algorithm. The ergodicity of the latter is revealed to be problematic, but this is overcome by combining both algorithms. Also, finite-size scaling is applied, returning a critical inverse temperature of \\epsilon \\beta = 0.79 plus or minus 0.02 compared to the known analytical value of log[2]. Furthermore, although the ice rules prevent local renormalization such as by Kadanoff block spins, a functioning coarse graining prescription based on the height function is proposed. This turns out to renormalise the high temperature states into states with temperatures below the critical point, thus preventing the determination of critical exponents from the renormalization flow.
We report two sets of high-precision Monte Carlo simulations of the two-dimensional XY model in Villain's formulation on large square lattices, employing the single-cluster update algorithm. In one set of simulations we use improved estimators to study the correlation length \ensuremath{\xi} and susceptibility \ensuremath{\chi} in the high-temperature phase. On a 1200\ifmmode\times\else\texttimes\fi{}1200 lattice this allows measurements up to \ensuremath{\xi}\ensuremath{\approxeq}140 with statistical errors less than 0.25%. We judge quantitatively the advantage of using improved estimators, estimate autocorrelation times, and compare the numerical efficiency of various definitions of the correlation length. From least-square fits to these data we find clear support for the exponential divergence predicted by Kosterlitz and Thouless. The other set of simulations is performed in the vicinity of the transition point. Here we apply finite-size-scaling theory to obtain an estimate for the exponent \ensuremath{\eta} at criticality.
Using the x-y model and a nonlocal updating scheme called cluster Monte Carlo, we calculate the superfluid density of a two-dimensional superfluid on large-size square lattices L\ifmmode\times\else\texttimes\fi{}L up to 400\ifmmode\times\else\texttimes\fi{}400. This technique allows us to approach temperatures close to the critical point, and by studying a wide range of L values and applying finite-size scaling theory we are able to extract the critical properties of the system. We calculate the superfluid density and from that we extract the renormalization-group beta function. We derive finite-size scaling expressions using the Kosterlitz-Thouless-Nelson renormalization group equations and show that they are in very good agreement with our numerical results. This allows us to extrapolate our results to the infinite-size limit. We also find that the universal discontinuity of the superfluid density at the critical temperature is in very good agreement with the Kosterlitz-Thouless-Nelson calculation and experiments.