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Progress of Theoretical Physics

Crossover Exponent of the Spin Anisotropic n-Vector Model with Short-Range Interaction in 1/n Expansion

S. Hikami, R. Abe

Published Aug 1, 1974• 24 citations
Publisher

Abstract

The critical behavior of a spin anisotropic n-vector model is studied in 1/n expansion. The critical exponent for the anisotropy (the crossover exponent) is derived in powers of 1/n to first order by considering the shift of critical temperature and by studying the derivatives of the susceptibility with respect to the anisotropic parameter. It is shown that obtained crossover exponent φ= 2/(d-2) + 4γ1/3n + O(n-2) is consistent with the result of Wilson's ε expansion.

DOI
10.1143/ptp.52.369

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Progress of Theoretical Physics•Jun 1, 1976•16 citations
The critical behavior of a spin anisotropic n-vector model is studied in 1/n expansion. The critical exponent for the anisotropy (the crossover exponent) is derived in powers of 1/n to first order by considering the shift of critical temperature and by studying the derivatives of the susceptibility with respect to the anisotropic parameter. It is shown that obtained crossover exponent φ= 2/(d-2) + 4γ1/3n + O(n-2) is consistent with the result of Wilson's ε expansion.
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 $\ensuremath{\gamma}\ensuremath{\simeq}1.38$).
Abstract Firstly the renormalization group technique is applied to an isotropic Heisenberg spin system with long‐range interaction and a set of critical exponents are calculated. These exponents are expressed in terms of a potential parameter s̀. All critical exponents except β, the magnetization exponent, depend on s̀. Secondly, the anisotropy is considered and the related crossover exponent φ is evaluated. This later parameter turns out to be equal to γ, the susceptibility exponent, as predicted by some authors.
Critical behavior in anisotropic cubic systems with the short-range interaction is studied by means of the Callan-Symanzik equations. As the static critical behavior, the stability of fixed points, the critical exponents ηC, γC, φsC and φcC, etc. are investigated. As the dynamic critical behavior, the dynamic critical exponent zφ is derived on the basis of the time dependent Ginzburg-Landau stochastic model. New results are a correction term of order ε3 in ηC, the crossover index φsC and the dynamic critical exponent zφ.