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arXiv:cond-mat/0007289 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Finite size scaling in the two-dimensional XY model and generalized universality

Guillermo Palma, T. Meyer, Raúl Labbé

Published Aug 16, 2002• 15 citations
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Abstract

In recent works [S. T. Bramwell, P. C. W. Holdsworth, and J.-F. Pinton, Nature (London) 396, 552 (1998); S. T. Bramwell et al., Phys. Rev. Lett. 84, 3744 (2000)], a generalized universality has been proposed, linking phenomena as dissimilar as two-dimensional (2D) magnetism and turbulence. To test these ideas, we performed Monte Carlo simulations of the 2D XY model. We found that the shape of the probability distribution function for the magnetization M is non-Gaussian and independent of the system size-in the range of the lattice sizes studied-below the Kosterlitz-Thoules temperature. However, our results suggest that in the full 2D XY model the shape of these distributions has a slight dependence on temperature-for finite volume-below the lattice-shifted critical temperature T*(L). This behavior can be explained by using renormalization group arguments and an extended finite-size scaling analysis, and by the existence of bounds for M.

DOI
10.1103/physreve.66.026108

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In recent works [S. T. Bramwell, P. C. W. Holdsworth, and J.-F. Pinton, Nature (London) 396, 552 (1998); S. T. Bramwell et al., Phys. Rev. Lett. 84, 3744 (2000)], a generalized universality has been proposed, linking phenomena as dissimilar as two-dimensional (2D) magnetism and turbulence. To test these ideas, we performed Monte Carlo simulations of the 2D XY model. We found that the shape of the probability distribution function for the magnetization M is non-Gaussian and independent of the system size-in the range of the lattice sizes studied-below the Kosterlitz-Thoules temperature. However, our results suggest that in the full 2D XY model the shape of these distributions has a slight dependence on temperature-for finite volume-below the lattice-shifted critical temperature T*(L). This behavior can be explained by using renormalization group arguments and an extended finite-size scaling analysis, and by the existence of bounds for M.
In recents works [1] (BHP), a generalized universality has been proposed, linking phenomena as dissimilar as 2D magnetism and turbulence. This universal behavior was attributed to systems sharing the the properties of high correlations, self-organized criticality and finite size effects. To study these phenomena, we performed a high precision Monte Carlo (MC) simulation of the 2DXY-model [2]. We found that the shape of the probability distribution function for the magnetization M is non Gaussian and independent of the system size, below the Kosterlitz-Thoules temperature [3]. However, the shape of these distributions do depend on the temperature, contrarily to the claim of (BHP). This behaviour is succesfully explained by using an extended finite-size scaling analysis and the existence of bounds for M.
In this paper we propose a method to study critical systems numerically, which combines collective-mode algorithms and renormalization group on the lattice. This method is an improved version of the Monte Carlo renormalization group in the sense that it has all the advantages of cluster algorithms. As an application we considered the 2D Ising model and studied whether scale invariance or universality are possible underlying mechanisms responsible for the approximate "universal fluctuations" close to a so-called bulk temperature T(L) . "Universal fluctuations" were first proposed in the work of Bramwell, Holdsworth, and Pinton [Nature (London) 396, 552 (1998)] and stated that the probability density function of a global quantity for very dissimilar systems, such as a confined turbulent flow and a two-dimensional (2D) magnetic system, properly normalized to the first two moments, becomes similar to the "universal distribution," originally obtained for magnetization in the 2D XY model in the low-temperature region. The results for the critical exponents and the renormalization-group flow of the probability density function are very accurate and show no evidence to support that the approximate common shape of the PDF should be related to both scale invariance or universal behavior.
The thermodynamics of the two-dimensional (2D) XY model is formulated by a transfer-matrix method and analyzed by a density-matrix renormalization group. The finite-size scaling and the beta function of the model are studied by the Roomany-Wyld renormalization-group theory. It is found that the 2D XY model has an essential finite-size effect and the Berezinskii-Kosterlitz-Thouless transition with the critical temperature TBKT=0.892{T}_{\mathrm{BKT}}=0.892TBKT​=0.892 appears in a finite system of 2000--3000 spins as a massless to massive transition with the effective critical temperature ${T}_{c}=1.07\ifmmode\pm\else\textpm\fi{}0.01.$
We study the two-dimensional generalized XY model that depends on an integer qqq by the Monte Carlo method. This model was recently proposed by Romano and Zagrebnov. We find a single Kosterlitz-Thouless (KT) transition for all values of qqq, in contrast with the previous speculation that there may be two transitions, one a regular KT transition and another a first-order transition at a higher temperature. We show the universality of the KT transitions by comparing the universal finite-size scaling behaviors at different values of qqq without assuming a specific universal form in terms of the KT transition temperature TKTT_{\rm KT}TKT​.
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.