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Journal of Physics C Solid State Physics

Universality and crossover in an extremely anisotropic n-vector model

Amnon Aharony A

Published Feb 21, 1974• 10 citations
Publisher

Abstract

The critical behaviour of an anisotropic n-vector model with pure single-component or Ising-like coupling is studied. The non-interacting spin components are integrated over, and the resulting single-component, reduced hamiltonian represents the gaussian model for n= infinity and the Ising-Ginzburg-Landau-Wilson model for finite n. For large n one finds T c (n)-T c ( infinity ) varies as 1/n, while the susceptibility behaves as chi approximately t -1 X(1/nt phi ), with t=T/T c (n)-1 and phi = 1 / 2 (4-d).

DOI
10.1088/0022-3719/7/4/001

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The critical behaviour of an anisotropic n-vector model with pure single-component or Ising-like coupling is studied. The non-interacting spin components are integrated over, and the resulting single-component, reduced hamiltonian represents the gaussian model for n= infinity and the Ising-Ginzburg-Landau-Wilson model for finite n. For large n one finds T c (n)-T c ( infinity ) varies as 1/n, while the susceptibility behaves as chi approximately t -1 X(1/nt phi ), with t=T/T c (n)-1 and phi = 1 / 2 (4-d).
The critical behaviour of an anisotropic n-vector model with purely Ising-like coupling is studied on the FCC lattice by means of high-temperature series. The model includes the s= 1 / 2 (n=1) and s= infinity (n=3) Ising models as special cases and exhibits rigorously mean-field behaviour at n= infinity . The crossover region 1/n to 0, t identical to (1-v/v c ) to 0, is examined in detail. The susceptibility exponent gamma remains Ising-like for all finite n. The critical temperature and critical susceptibility amplitude vary as 1/n.
We investigate a two-dimensional Ising model with long-range interactions that emerge from a generalization of the magnetic dipolar interaction in spin systems with in-plane spin orientation. This interaction is, in general, anisotropic whereby in the present work we focus on the isotropic case for which the model is found to be at its upper critical dimensionality. To investigate the critical behavior the temperature and field dependence of several quantities are studied by means of Monte Carlo simulations. On the basis of the Privman-Fisher hypothesis and results of the renormalization group the numerical data are analyzed in the framework of a finite-size scaling analysis and compared to finite-size scaling functions derived from a Ginzburg-Landau-Wilson model in zero mode (mean-field) approximation. The obtained excellent agreement suggests that at least in the present case the concept of universal finite-size scaling functions can be extended to the upper critical dimensionality.
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