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Physical Review Letters

Order-Parameter Distribution Function of Finite O(n) Symmetric Systems

Chen Xs, V. Dohm, Norbert Schultka

Published Oct 21, 1996• 39 citations
Publisher

Abstract

We present analytic and numerical studies of the order-parameter distribution function near the critical point of $O(n)$ symmetric three-dimensional (3D) systems in a finite geometry. The distribution function is calculated within the ${\ensuremath{\phi}}^{4}$ field theory for a 3D cube with periodic boundary conditions by means of a new approach that appropriately deals with the Goldstone modes below ${T}_{c}$. Good agreement is found with new Monte Carlo data for the distribution function of the magnetization of the 3D $\mathrm{XY}$ $(n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2)$ and Heisenberg $(n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3)$ models.

DOI
10.1103/physrevlett.77.3641

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We present a renormalization-group study of the order-parameter distribution function near the critical point of O(n) symmetric three-dimensional (3D) systems in a finite geometry. The distribution function is calculated within the φ4 field theory for a 3D cube with periodic boundary conditions by means of a novel approach that appropriately deals with the Goldstone modes below T c . Results are given for both vanishing and finite external field h. The results describe finite-size effects near the critical point in the h– T-plane including the first-order transition at the coexistence line at h = 0 below T c . Quantitative theoretical predictions of the finite-size scaling function are presented for the Ising (n=1), XY(n=2) and Heisenberg (n=3) models. Good agreement is found with recent Monte Carlo data.
We present field-theoretic and numerical studies of finite-size effects on the exponential relaxation times ${\ensuremath{\tau}}_{1}$ and ${\ensuremath{\tau}}_{2}$ of the order parameter and the square of the order parameter near the critical point of three-dimensional Ising-like systems. Finite-size scaling functions of ${\ensuremath{\tau}}_{1}$ and ${\ensuremath{\tau}}_{2}$ are calculated for the relaxational dynamics of the ${\ensuremath{\phi}}^{4}$ model in cubic geometry with periodic boundary conditions. At Tc{T}_{c}Tc​ we predict the universal ratio $\frac{{\ensuremath{\tau}}_{1}}{{\ensuremath{\tau}}_{2}}=5.4$. Below Tc{T}_{c}Tc​, a maximum of ${\ensuremath{\tau}}_{2}$ is predicted. New Monte Carlo data for the Ising model are presented which are in good agreement with the predictions.
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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).
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