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Journal of Physics A Mathematical and General

Magnetization fluctuations in a ferromagnet with mean-field interactions on a slab

N Angelescu, M. Bundaru

Published Mar 7, 1994• 1 citation
Publisher

Abstract

The spin probability distribution at equilibrium in a multicomponent mean-field model of a ferromagnetic slab made of N rows is considered. Everywhere, but at the critical point, the spin fluctuations converge, as the number of spins pet row tends to infinity , to Gaussians with covariance equal to the thermodynamic susceptibility matrix. At criticality, abnormal coherent fluctuations along the positive eigenvector of the interaction matrix are found, whose limit distribution is an exp(-x 4 ) law.

DOI
10.1088/0305-4470/27/5/013

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The spin probability distribution at equilibrium in a multicomponent mean-field model of a ferromagnetic slab made of N rows is considered. Everywhere, but at the critical point, the spin fluctuations converge, as the number of spins pet row tends to infinity , to Gaussians with covariance equal to the thermodynamic susceptibility matrix. At criticality, abnormal coherent fluctuations along the positive eigenvector of the interaction matrix are found, whose limit distribution is an exp(-x 4 ) law.
The equilibrium magnetization distribution for an Ising-spin model in the mean-field approximation is used to present a simple study of the ferromagnetic phase transition in the absence of external magnetic field (spontaneous magnetization). For temperatures below a critical one (the Curie Temperature) this distribution exhibits two peaks, showing that two states of magnetization are possible. At the Curie temperature the distribution presents a plateau, showing large fluctuations in the magnetization. A scaling-like analysis of the distribution shows that the Curie point can be characterized as an unstable fixed point of a recursion relation.
The authors consider the mean-field version of a spin-1 Ising ferromagnet in a random crystal field described by a Gaussian probability distribution. Depending on the width of the Gaussian, they obtain a rich phase diagram, with critical and coexistence lines and some multicritical points. At low temperatures, their numerical results are supported by some analytic asymptotic expansions. They also calculate the ground state for a suitable two-valued delta-function distribution to compare with the results for the Gaussian case.
We introduce a one dimensional Ising model with two competing interactions: nearest neighbor random couplings +/-J with equal probability and a positive infinite range coupling Lambda. At low temperature T the model exhibits a first order phase transition between a ferromagnetic state (with magnetization m(1) = 1 at T = 0) and a << ferrimagnetic >> state (with m(2) = 2/3 at T = 0), when the disorder strength J/Lambda is increased. For 5/12 < J/Lambda < 1, a whole spectrum of ferrimagnetic ground states with magnetization m(n) = 2/(n + 1) (n = 2, ..., infinity) is present while for J/lambda > 1 the ground state is given by a trivial one dimensional spin glass with m = 0. The main qualitative features of the model can be described by a simplified annealed model where the random couplings can arrange themselves to minimize free energy with the constraint that the number of positive couplings is fixed by the law of large numbers in the thermodynamic limit. This model is exactly solved at all temperatures and the diagram of phase is calculated.
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.
We present new numerical simulations of the random field Ising model in three dimensions at zero temperature. The critical exponents are found to agree with previous results. We study the magnetic susceptibility by applying a small magnetic field perturbation. We find the critical amplitude ratio of the magnetic susceptibilities to be very large, equal to 233.1 ± 1.5. We find strong sample to sample fluctuations which obey finite size scaling. The probability distribution of the size of small energy excitations is maximally non-self-averaging, obeying a double peak distribution, and is finite size scaling invariant. We also study the approach to the thermodynamic limit of the ground state magnetization at the phase transition.
Although the fully connected Ising model does not have a length scale, we show that the critical exponents for thermodynamic quantities such as the mean magnetization and the susceptibility can be obtained using finite size scaling with the scaling variable equal to N, the number of spins. Surprisingly, the mean value and the most probable value of the magnetization are found to scale differently with N at the critical temperature of the infinite system, and the magnetization probability distribution is not a Gaussian, even for large N. Similar results inconsistent with the usual understanding of mean-field theory are found at the spinodal. We relate these results to the breakdown of hyperscaling and show that hyperscaling can be restored by increasing N while holding the Ginzburg parameter rather than the temperature fixed, or by doing finite size scaling at the pseudocritical temperature where the susceptibility is a maximum for a given value of N. We conclude that finite size scaling for the fully connected Ising model yields different results depending on how the mean-field limit is approached.