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Surface properties at the Kosterlitz-Thouless transition

Bertrand Berche, Université Henri Poincaré

Published Jan 1, 2008• 2 citations
OpenAlex

Abstract

Monte Carlo simulations of the two-dimensional XY model are performed in a square geometry with free and mixed fixed-free boundary conditions. Using a Schwarz-Christoffel conformal mapping, we deduce the exponent η of the order parameter correlation function and its surface equivalent η ‖ at the Kosterlitz-Thouless transition temperature. The well known value η(TKT) = 1/4 is easily recovered even with systems of relatively small sizes, since the shape effects are encoded in the conformal mapping. The exponent associated to the surface correlations is similarly obtained η1(TKT) ≃ 0.54. 1

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Monte Carlo simulations of the two-dimensional XY model are performed in a square geometry with various boundary conditions (BC). Using conformal mappings we deduce the exponent η σ ( T ) of the order parameter correlation function and its surface analogue η ∥ ( T ) as a function of the temperature in the critical (low-temperature) phase of the model. The temperature dependence of both exponents is obtained numerically with a good accuracy up to the Kosterlitz–Thouless transition temperature. The bulk exponent follows from simulations of correlation functions with periodic boundary conditions or order parameter profiles with open boundary conditions and with symmetry breaking surface fields. At very low temperatures, η σ ( T ) is found in pretty good agreement with the linear temperature-dependence of Berezinskii's spin-wave approximation. We also show some evidence that there are no noticeable logarithmic corrections to the behaviour of the order parameter density profile at the Kosterlitz–Thouless (KT) transition temperature, while these corrections exist for the correlation function. At the KT transition the value η σ ( T KT ) = 1/4 is accurately recovered. The exponent associated with the surface correlations is similarly obtained after a slight modification of the boundary conditions: the correlation function is computed with free BC, and the profile with mixed fixed–free BC. It exhibits a monotonic behaviour with temperature, starting linearly according to the spin-wave approximation and increasing up to a value η ∥ ( T KT ) ≃ 1/2 at the Kosterlitz–Thouless transition temperature. The thermal exponent η ε ( T ) is also computed and we give some evidence that it keeps a constant value in agreement with the marginality condition of the temperature field below the KT transition.

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Monte Carlo simulations of the two-dimensional XY model are performed in a square geometry with free and mixed fixed-free boundary conditions. Using a Schwarz-Christoffel conformal mapping, we deduce the exponent η of the order parameter correlation function and its surface equivalent η ‖ at the Kosterlitz-Thouless transition temperature. The well known value η(TKT) = 1/4 is easily recovered even with systems of relatively small sizes, since the shape effects are encoded in the conformal mapping. The exponent associated to the surface correlations is similarly obtained η1(TKT) ≃ 0.54. 1
Monte Carlo simulations of the two-dimensional XY model are performed in a square geometry with various boundary conditions (BC). Using conformal mappings we deduce the exponent η σ ( T ) of the order parameter correlation function and its surface analogue η ∥ ( T ) as a function of the temperature in the critical (low-temperature) phase of the model. The temperature dependence of both exponents is obtained numerically with a good accuracy up to the Kosterlitz–Thouless transition temperature. The bulk exponent follows from simulations of correlation functions with periodic boundary conditions or order parameter profiles with open boundary conditions and with symmetry breaking surface fields. At very low temperatures, η σ ( T ) is found in pretty good agreement with the linear temperature-dependence of Berezinskii's spin-wave approximation. We also show some evidence that there are no noticeable logarithmic corrections to the behaviour of the order parameter density profile at the Kosterlitz–Thouless (KT) transition temperature, while these corrections exist for the correlation function. At the KT transition the value η σ ( T KT ) = 1/4 is accurately recovered. The exponent associated with the surface correlations is similarly obtained after a slight modification of the boundary conditions: the correlation function is computed with free BC, and the profile with mixed fixed–free BC. It exhibits a monotonic behaviour with temperature, starting linearly according to the spin-wave approximation and increasing up to a value η ∥ ( T KT ) ≃ 1/2 at the Kosterlitz–Thouless transition temperature. The thermal exponent η ε ( T ) is also computed and we give some evidence that it keeps a constant value in agreement with the marginality condition of the temperature field below the KT transition.
Monte Carlo simulations of the two-dimensional XY model are performed in a square geometry with fixed boundary conditions. Using a conformal mapping, it is very easy to deduce the exponent η σ ( T ) of the order parameter correlation function at any temperature in the critical phase of the model. The temperature behaviour of η σ ( T ) is obtained numerically with a good accuracy up to the Kosterlitz-Thouless transition temperature. At very low temperatures, a good agreement is found with Berezinskii's harmonic approximation. Surprisingly, we show some evidence that there are no logarithmic corrections to the behaviour of the order parameter density profile (with symmetry-breaking surface fields) at the Kosterlitz-Thouless transition temperature.
We present detailed Monte Carlo results for the susceptibility \ensuremath{\chi}, correlation length \ensuremath{\xi}, and specific heat Cv{\mathit{C}}_{\mathit{v}}Cv​ for the XY model. The simulations are done on 642{64}^{2}642, 1282{128}^{2}1282, 2562{256}^{2}2562, and 5122{512}^{2}5122 lattices over the temperature range 0.98\ensuremath{\le}T\ensuremath{\le}1.43 corresponding to 270. Fits to \ensuremath{\chi} and \ensuremath{\xi} data favor a Kosterlitz-Thouless (KT) singularity over a second-order transition; however, unconstrained four-parameter KT fits do not confirm the predicted values \ensuremath{\nu}=0.5 and \ensuremath{\eta}=0.25. Our best estimate, Tc{\mathit{T}}_{\mathit{c}}Tc​=0.894(5), is obtained using KT fits with \ensuremath{\nu} fixed at 0.5. The exponent \ensuremath{\eta} is calculated as a function of temperature in the spin-wave phase using Monte Carlo renormalization-group and finite-size-scaling methods. Both methods give consistent results and we find \ensuremath{\eta}\ensuremath{\approxeq}0.235 at T=0.894. We also present results for the behavior of vortex density across the transition and exhibit how the dilute-gas approximation breaks down.
With a large-scale Monte Carlo simulation, non-equilibrium dynamics of the two-dimensional fully frustrated XY model is investigated. We tackle the Kosterlitz-Thouless phase transition. Starting from an ordered initial state, we study the dynamic evolution of the magnetization as well as a specifically defined Binder cumulant. From the dynamic scaling ansatz, we extract the correlating time of the dynamics and the spatial correlation length of the equilibrium state. The dynamic exponent z is determined with relatively high accuracy. Especially, we suggest and demonstrate how one may directly measure the dynamic exponent z above TKT from the scaling fit of the Binder cumulant. These results indicate that the dynamic exponent z fluctuates around z=2, and this is consistent with that observed at temperatures below the transition temperature TKT.
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.