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

Abstract

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...

Finite-size corrections to scaling of the magnetization distribution in the two-dimensionalXYmodel at zero temperature

“…The probability density function (PDF) of the magnetization, P L (m), has been much studied [9,16,[18][19][20][21][22]24]. Its zero temperature limit (T → 0) has been first calculated [24] by using the spin-wave approximation cos…”

“…In the majority of the examples, however, the experiments or simulations provide data of insufficient accuracy to decide unambiguously about the universality class. Indeed, it is not easy even to observe the changes occurring in P L (m) if the XY model is considered at finite temperatures where the spin-wave approximation breaks down [20][21][22].…”

Origin of the approximate universality of distributions in equilibrium correlated systems

“…These variations can be large, coming from non-linear instanton-like objects, taking the distribution far from the generic asymmetric form shown in Fig.1(b) [8,16], or they can be smaller, coming from the harmonic contribution itself. That is, the BHP function is not a miraculous unique function for all circumstances, rather distributions of the form of the first term in equation (7) depend weakly on the boundary conditions [6,17,18], on the mass [14] and even on the temperature [9,10]. Of these parameters the strongest dependence is on dimension: we expect that 3D correlated equilibrium systems are related to the 3D Gaussian model, which has weakly asymmetric order parameter fluctuations and is not critical [6,19].…”

“…In fact it is easy to find critical systems where the distribution is radically different and in most situations, some deviation from BHP distribution is apparent. Even in the case of the 2D-XY model itself it has been recently established that small temperature-dependent corrections exist [7,9,10]. There are therefore strict physical criteria associated with both the observation of this generic behaviour and with the deviations from it, just as in the case of Gaussian fluctuations through the application of the central limit theorem.…”

“…The second term comes from the contribution of instantons that restore phase space symmetry and so is strongly dependent on the universality class. It is the analogue of the corrections computed exactly for the 2D-XY model within the spin wave approximation [9,10] The existence of a point, or points, in the phase diagram where the PDF of the 2D Ising model is close to the BHP distribution [8] results from a compromise between these two terms: to be as close as possible to BHP, we have to reduce the mass in the Gaussian propagators of the first term. This is exactly what happens as one approaches T c and the instantaneous value of φ 0 reduces.…”