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Revista Mexicana de Física

Universalidad generalizada en sistemas de tamaño finito altamente correlacionados

Guillermo Palma, T. I. Meyer, Raúl Labbé

Published Nov 1, 2002• 0 citations
OpenAlex

Abstract

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.

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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 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 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.
This thesis addresses the nature of global (many body) fluctuations in highly correlated systems. We begin with the question of temperature dependence in finite two dimensional XY (2dXY) model magnets. Such systems have a fully critical low temperature phase. It is shown analytically, backed up by extensive Monte Carlo simulations, that the non-Gaussian distribution of order parameter fluctuations is not strictly universal but has an explicit temperature dependence - contrary to previous findings. The temperature dependence is used to explain why past studies derived the same distribution for fluctuations of the full order parameter and an approximate linearized form. The appearance of spin vortices in the related Harmonic model is discussed and an argument is presented for why these defects must always appear as bound pairs. The linearized order parameter of the 2dXY model leads to a family of dimensionally dependent models defined in reciprocal space. An argument is presented for the interpretation of these systems as being critical and a direct space Hamiltonian is derived for the one dimensional case. This model has order parameter fluctuations distributed according to the Fisher-Tippett-Gumbel distribution from extreme value statistics (EVS). The link between criticality and EVS is investigated, as are the origins of the non-Gaussianity. The ability to distinguish between critical distributions is discussed. It is seen that for one, two and three dimensions the critical models presented lead to functionally similar fluctuation distributions. A previously reported link between EVS and l/f noise is investigated. Our one dimensional critical model is mapped onto the action used to generate l/f signals and we propose an alternative interpretation of the link in the context of a 1/q dispersion of spatial normal modes. The experimental observability of the FTG distribution in l/f signals is considered with emphasis on imperfections in the noise. A physically relevant method of generating l/f noise from the superposition of random telegraph signals is also examined.
The magnetization probability density in d = 2 and 3 dimensional Ising models in slab geometry of volume [Formula: see text] is computed through Monte-Carlo simulation at the critical temperature and zero magnetic field. The finite-size scaling of this distribution and its dependence on the system aspect-ratio [Formula: see text]and boundary conditions are discussed. In the limiting case [Formula: see text] of a macroscopically large slab ([Formula: see text]) the distribution is found to scale as a Gaussian function for all tested system sizes and boundary conditions.
Using large-scale Monte Carlo simulations that combine parallel tempering with specialized cluster updates, we show that Ising spin glasses with L\'evy-distributed interactions share the same universality class as Ising spin glasses with Gaussian or bimodal-distributed interactions. Corrections to scaling are large for L\'evy spin glasses. To overcome these and show that the critical exponents agree with the bimodal and Gaussian case, we perform an extended scaling of the two-point finite-size correlation length and the spin-glass susceptibility. Furthermore, we compute the critical temperature and compare its dependence on the disorder distribution width with recent analytical predictions [J. Stat. Mech. (2008) P04006].
We study the probability distribution P(M) of the order parameter (average magnetization) M, for the finite-size systems at the critical point. The systems under consideration are the 3-dimensional Ising model on a simple cubic lattice, and its 3-state generalization known to have remarkably small corrections to scaling. Both models are studied in a cubic box with periodic boundary conditions. The model with reduced corrections to scaling makes it possible to determine P(M) with unprecedented precision. We also obtain a simple, but remarkably accurate, approximate formula describing the universal shape of P(M).