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Progress of Theoretical Physics

Critical Behavior of the Anisotropic Classicaln-Vector Model

Masuo Suzuki

Published May 1, 1973• 12 citations
Publisher

Abstract

The critical exponents of the anisotropic classical model with n components are investigated in the 1/n-expansion. It is found that they depend only upon the number of the spin components with the largest interaction. This gives a partial confirmation of the universality. It is also proved that the singular part of the derivative of the perpendicular susceptibility is proportional to the dominant singularity of the specific heat through the first order of 1/n: i.e., γ⊥=α-1+O(1/n2).

DOI
10.1143/ptp.49.1451

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Progress of Theoretical Physics•May 1, 1973•17 citations
The critical exponents of the anisotropic classical model with n components are investigated in the 1/n-expansion. It is found that they depend only upon the number of the spin components with the largest interaction. This gives a partial confirmation of the universality. It is also proved that the singular part of the derivative of the perpendicular susceptibility is proportional to the dominant singularity of the specific heat through the first order of 1/n: i.e., γ⊥=α-1+O(1/n2).
The critical behavior of a system of nnn-component classical "spins"with anisotropic pair interactions is discussed using renormalization-group techniques for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$. To first order in $\ensuremath{\epsilon}$ the correlation and susceptibility exponents in the isotropic limit are $2\ensuremath{\nu}=\ensuremath{\gamma}=1+[\frac{(n+2)}{2(n+8)}]\ensuremath{\epsilon}$, while the anisotropy or crossover exponent is $\ensuremath{\varphi}=1+[\frac{n}{2(n+8)}]\ensuremath{\epsilon}$. When $n\ensuremath{\rightarrow}\ensuremath{\infty}$ these expansions agree with exact spherical-model results. For n=3n=3n=3 and d=3d=3d=3 series expansions indicate $\ensuremath{\varphi}\ensuremath{\simeq}1.2$ (compared with $\ensuremath{\gamma}\ensuremath{\simeq}1.38$).
For pt.I see J. Phys. C: Solid St. Phys., vol.6, p.2691 (1973). The critical behaviour of the step model of phase transitions on the square, simple cubic and body-centred cubic lattices is investigated by the method of exact series expansions. The model is found to be in agreement with two aspects of universality, namely the dependence of critical exponents on spin-space dimensionality and the lattice independence of critical exponents for fixed lattice and spin-space dimensionality. Series expansions for the three-dimensional susceptibility and specific heats are generated and analysed, and the critical exponents are found to be gamma =1.335+or-0.01 and alpha =-0.04+or-0.10 respectively. In two dimensions the existence and nature of a phase transition remains an open question. No evidence of either a conventional algebraic singularity or a vortex induced essential singularity is found.