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arXiv:cond-mat/0102283 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Universal magnetic fluctuations with a field-induced length scale

Baptiste Portelli, P. C. W. Holdsworth, Mauro Sellitto, S. T. Bramwell

Published Aug 27, 2001• 35 citations
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Abstract

We calculate the probability density function for the order-parameter fluctuations in the low-temperature phase of the two-dimensional $\mathrm{XY}$ model of magnetism near the line of critical points. A finite correlation length $\ensuremath{\xi},$ is introduced with a small magnetic field h, and an expression for $\ensuremath{\xi}(h)$ is developed by treating nonlinear contributions to the field energy using a Hartree approximation. We find analytically a series of universal non-Gaussian distributions of the finite-size scaling form $P(m,L,\ensuremath{\xi})\ensuremath{\sim}{L}^{\ensuremath{\beta}/\ensuremath{\nu}}{P}_{L}{(mL}^{\ensuremath{\beta}/\ensuremath{\nu}},\ensuremath{\xi}/L)$ and present a function of the form $P(x)\ensuremath{\sim}{\mathrm{exp}[x\ensuremath{-}\mathrm{exp}(x)]{}}^{a(h)}$ that gives the probability density functions to an excellent approximation. We propose $a(h)$ as an indirect measure of the length scale of correlations in a wide range of complex systems.

DOI
10.1103/physreve.64.036111

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We calculate the probability density function for the order-parameter fluctuations in the low-temperature phase of the two-dimensional XY\mathrm{XY}XY model of magnetism near the line of critical points. A finite correlation length \ensuremathξ,\ensuremath{\xi},\ensuremathξ, is introduced with a small magnetic field h, and an expression for \ensuremathξ(h)\ensuremath{\xi}(h)\ensuremathξ(h) is developed by treating nonlinear contributions to the field energy using a Hartree approximation. We find analytically a series of universal non-Gaussian distributions of the finite-size scaling form P(m,L,\ensuremathξ)\ensuremath∼L\ensuremathβ/\ensuremathνPL(mL\ensuremathβ/\ensuremathν,\ensuremathξ/L)P(m,L,\ensuremath{\xi})\ensuremath{\sim}{L}^{\ensuremath{\beta}/\ensuremath{\nu}}{P}_{L}{(mL}^{\ensuremath{\beta}/\ensuremath{\nu}},\ensuremath{\xi}/L)P(m,L,\ensuremathξ)\ensuremath∼L\ensuremathβ/\ensuremathνPL​(mL\ensuremathβ/\ensuremathν,\ensuremathξ/L) and present a function of the form P(x)\ensuremath∼exp[x\ensuremath−exp(x)]a(h)P(x)\ensuremath{\sim}{\mathrm{exp}[x\ensuremath{-}\mathrm{exp}(x)]{}}^{a(h)}P(x)\ensuremath∼exp[x\ensuremath−exp(x)]a(h) that gives the probability density functions to an excellent approximation. We propose a(h)a(h)a(h) as an indirect measure of the length scale of correlations in a wide range of complex systems.
We discuss the probability distribution function for the magnetic order parameter M, in the low temperature phase of the two-dimensional XY model. In this phase the system is critical over the whole range of temperature. The thermally averaged value of the order parameter 〈M〉, which is zero in the thermodynamic limit, has abnormally large finite size corrections. An exact result, within a spin wave calculation gives 〈M〉=(1/2N)T/8πJ, where J is the magnetic exchange constant and N the number of spins. We show, using Monte Carlo simulation, that the distribution function, Q(y−〈y〉), y=T−1LT/4πJM, is an asymmetric universal function. Using a diagramatic technique, we show that the asymmetry comes from three-spin and higher correlations. If only two-spin correlations are considered, the distribution is Gaussian. However, as there are contributions from two-spin terms separated by all distances, the distribution remains broad and is consistent with a divergent susceptibility.
Order parameter fluctuations for the two-dimensional Ising model in the region of the critical temperature are presented. A locus of temperatures T(*) (L) and a locus of magnetic fields B(*) (L) are identified, for which the probability density function is similar to that for the two-dimensional XY model in the spin wave approximation. The characteristics of the fluctuations along these points are largely independent of universality class. We show that the largest range of fluctuations relative to the variance of the distribution occurs along these loci of points, rather than at the critical temperature itself and we discuss this observation in terms of intermittency. Our motivation is the identification of a generic form for fluctuations in correlated systems in accordance with recent experimental and numerical observations. We conclude that a universality-class-dependent form for the fluctuations is a particularity of critical phenomena related to the change in symmetry at a phase transition.
Transfer-matrix methods are used to calculate spin-spin correlation functions (G), Helmholtz free energies (f) and magnetizations (m) in the two-dimensional random-field Ising model close to the zero-field bulk critical temperature T(c 0), on long strips of width L=3-18 sites, for binary field distributions. Analysis of the probability distributions of G for varying spin-spin distances R shows that describing the decay of their averaged values by effective correlation lengths is a valid procedure only for not very large R. Connections between field and correlation function distributions at high temperatures are established, yielding approximate analytical expressions for the latter, which are used for computation of the corresponding structure factor. It is shown that, for fixed R/L, the fractional widths of correlation-function distributions saturate asymptotically with L-2.2. Considering an added uniform applied field h, a connection between f(h), m(h), the Gibbs free energy g(m) and the distribution function for the uniform magnetization in a zero uniform field, P0(m), is derived and first illustrated for pure systems, and then applied for nonzero random field. From finite-size scaling and crossover arguments, coupled with numerical data, it is found that the width of P0(m) varies against (nonvanishing, but small) random-field intensity H0 as H(-3/7)(0).
We calculate the two-dimensional probability for the magnetization in a two-dimensional XY model of finite size. We show that, for arbitrarily large N, there is a topological difference between the distributions in the low-temperature spin wave regime and in the high-temperature paramagnetic regime. In the low-temperature phase is a well-defined ring function and we calculate an upper bound for . Even so, this is consistent with the susceptibility per spin, , being divergent. We show further that the distribution function Q(M) for the scalar magnetization has a universal form, scaling with the single variable , where L is the system size and J is the coupling constant. We show that has considerable structure, with two terms of order N, one due to the spin waves and the other due to vortices. This leads to a peak in at the Kosterlitz - Thouless - Berezinskii transition.
Order parameter fluctuations for the two dimensional Ising model in the region of the critical temperature are presented. A locus of temperatures T*(L) and of magnetic fields B*(L) are identified, for which the probability density function is similar to that for the 2D-XY model in the spin wave approximation.The characteristics of the fluctuations along these points are largely independent of universality class. We show that the largest range of fluctuations relative to the variance of the distribution occurs along these loci of points, rather than at the critical temperature itself and we discuss this observation in terms of intermittency. Our motivation is the identification of a generic form for fluctuations in correlated systems in accordance with recent experimental and numerical observations. We conclude that a universality class dependent form for the fluctuations is a particularity of critical phenomena related to the change in symmetry at a phase transition.
We present a detailed investigation of the probability density function (PDF) of order parameter fluctuations in the finite two-dimensional XY (2d XY ) 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 2d XY model: the harmonic and ‘harmonic XY ’ models. The harmonic model has no Kosterlitz–Thouless–Berezinskiĭ (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.