Finite size scaling in the two-dimensional<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>XY</mml:mi></mml:math>model and generalized universality

Palma, G.; Meyer, T. I.; Labbé, R.

doi:10.1103/physreve.66.026108

Cited by 15 publications

(18 citation statements)

References 33 publications

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“…From a visual point of view the distribution looks very similar across the critical, vortex free, region. The temperature dependence established here is weaker than, but of the same magnitude as, that claimed by Palma et al, for the full XY model [33]. More work is needed to fully understand this difference.…”

Section: Resultsmentioning

confidence: 44%

Temperature-dependent fluctuations in the two-dimensionalXYmodel

Banks¹,

Bramwell²

2005

J. Phys. A: Math. Gen.

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Abstract. We present a detailed investigation of the probability density function (PDF) of order parameter fluctuations in the finite two-dimensional XY (2dXY ) model. In the low temperature critical phase of this model, the PDF approaches a universal non-Gaussian limit distribution in the limit T → 0. Our analysis resolves the question of temperature dependence of the PDF in this regime, for which conflicting results have been reported. We show analytically that a weak temperature dependence results from the inclusion of multiple loop graphs in a previously-derived graphical expansion. This is confirmed by numerical simulations on two controlled approximations to the 2dXY model: the Harmonic and "Harmonic XY" models. The Harmonic model has no Kosterlitz-Thouless-Berezinskiȋ (KTB) transition and the PDF becomes progressively less skewed with increasing temperature until it closely approximates a Gaussian function above T ≈ 4π. Near to that temperature we find some evidence of a phase transition, although our observations appear to exclude a thermodynamic singularity.

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Section: Resultsmentioning

confidence: 44%

Temperature-dependent fluctuations in the two-dimensionalXYmodel

Banks¹,

Bramwell²

2005

J. Phys. A: Math. Gen.

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“…Nevertheless, more recently [10] a numerical study of the full 2D XY-model was performed, and the results suggested a small but systematic dependence of the PDF on the system temperature. This numerical analysis was performed by using high precision Monte Carlo simulations, which amounts to control the statistical independence of the configurations used to compute thermodynamical averages, as this system is affected by the well known critical slowing down effect.…”

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confidence: 99%

Corrections to universal fluctuations in correlated systems: The two-dimensionalXYmodel

2005

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Generalized universality, as recently proposed, postulates a universal non-Gaussian form of the probability density function (PDF) of certain global observables for a wide class of highly correlated systems of finite volume N . Studying the 2D XY -model, we link its validity to renormalization group properties. It would be valid if there were a single dimension 0 operator, but the actual existence of several such operators leads to T -dependent corrections. The PDF is the Fourier transform of the partition function Z(q) of an auxiliary theory which differs by a dimension 0 perturbation with a very small imaginary coefficient iq/N from a theory which is asymptotically free in the infrared. We compute the PDF from a systematic loop expansion of ln Z(q).PACS numbers: 05.70. Jk, 05.50.+q, 75.10.Hk, 68.35.Rh Trying to understand universality properties of critical systems near a second order phase transition was extremely fruitful for the development of physics. It lead to the development of Wilsons renormalization group (RG) [1] which found applications ranging from solid state physics to elementary particle theory. Universality classes in the RG-sense depend on the dimension of the system and on the symmetry properties of the order parameter.Recently a generalized universality has been proposed [2], which goes far beyond the known picture. It is therefore of great general interest to understand the conditions for its validity. Generalized universality is supposed to hold true for systems sharing the properties of strong correlations, finite volume and self similarity, no matter whether they have the same symmetries and dimensions nor whether they correspond to equilibrium or non-equilibrium systems. Such new universality is expressed in a non-Gaussian universal curve these systems share when the PDF of some global quantity of each system is plotted (universal fluctuations). Studying the 2D XY-model as an example, we link its validity to RGproperties of the system, viz. the existence of dimension 0 perturbations of a system which is asymptotically free in the infrared. It would be exactly true if there were one single dimension 0 operator. But the actual existence of several dimension 0 operators in the model leads to T-dependent corrections to the supposedly universal curve which do not go away when the volume N → ∞. * Electronic address: gerhard.mack@desy.de † Electronic address: gpalma@lauca.usach.cl ‡ Electronic address: lvergara@lauca.usach.cl, If there were no dimension 0 operator, the PDF would be Gaussian.Let us recall some previous results. The supposed universal curve for the PDF, which we call BHP, is nonGaussian in spite of the result that a naive application of the central limit theorem would have given; this possibility has been attributed [2] to the anomalous statistical property of finite size critical systems that, whatever their size is, cannot be divided into mesoscopic regions that are statistically independent, a necessary condition for the central limit theorem to apply. In Ref. [3] it was s...

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“…The former have been discussed in detail in Ref. [5,21]; at fixed temperature, they give small deviations from the universal curve for small system size, L. These are finite size corrections to the thermodynamic limit function and disappear with relatively modest increase in the system size.…”

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Vortex corrections to universal scaling of magnetic fluctuations in 2D XY model

Holdsworth

Sellitto

2002

Physica A: Statistical Mechanics and its Applications

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Abstract. The vortex contribution to the probability density function of longitudinal magnetization fluctuations is examined in finite 2D XY systems close to the Kosterlitz-Thouless-Berezinskii transition temperature. Within the temperature range studied their relevance is limited to rare fluctuations, where they increase the probability of events exceeding four standard deviations below the mean magnetization.The characterization of fluctuation statistics is a central problem in the study of critical phenomena, as the break down of Landau theory on approaching the critical point, implies a non-Gaussian distribution for order parameter fluctuations [1]. From renormalization group theory it is customary to think of critical phenomena divided into universality classes, characterized by the symmetry group of the order parameter and the spatial dimension. One would therefore expect critical fluctuation statistics to be determined essentially by the set of critical exponents (say β and ν for a regular critical point) which describe the scaling behavior of derivatives of the singular part of the free energy. Evidence from, e.g. the Ising model [2,3] and Potts models [4] suggests that this is indeed generally the case.We have, however recently considered an exception to this established phenomenology, the 2D XY model [5,6]. Here the low temperature phase consists of a line of critical points, with one independent critical exponent η(T ) = T /2πJ (with J the exchange constant) extending down to zero temperature and separated from the high temperature paramagnetic phase by the Kosterlitz-Thouless-Berezinskii (KTB) phase transition [7,8,9]. It is well established that for all temperatures below the KTB transition temperature, T KTB , renormalisation group flows are to a quadratic effective Hamiltonian [10,11], with the result that the asymptotic behaviour, in the limit of large system size is perfectly captured by a harmonic model. The advantage of such a simple model is that the probability density function, P (m), for fluctuations of the order parameter, m, can be calculated analytically, without using either renormalisation group or the scaling hypothesis. Surprisingly, we find that the form of distribution is independent of η along the whole line of critical points. This universal scaling function, plotted as the solid line in Fig. 1, arises when σP (m) is plotted against µ = (m − m )/σ, where m and σ are the mean and standard ‡ pcwh@ens-lyon.fr § sellitto@ictp.trieste.it

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Finite size scaling in the two-dimensionalXYmodel and generalized universality

Cited by 15 publications

References 33 publications

Temperature-dependent fluctuations in the two-dimensionalXYmodel

Temperature-dependent fluctuations in the two-dimensionalXYmodel

Corrections to universal fluctuations in correlated systems: The two-dimensionalXYmodel

Vortex corrections to universal scaling of magnetic fluctuations in 2D XY model

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