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Progress of Theoretical Physics

Critical Behavior in Anisotropic Cubic Systems with Short-Range Interaction

Y. Yamazaki

Published Jun 1, 1976• 16 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.55.1733

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Physical review. B, Condensed matter•Jan 26, 2001•9 citations
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.
Effects of the potential range on critical behavior in the second order phase transition are studied for isotropic N-component spin systems with long-range interactions by means of the Callan-Symanzik or renormalization-group equations. Critical exponents η, γ, φ, ν, α, δ and β and the scaling laws are derived. The equation of state, the correctional index ω and the logarithmic corrections in 2σ dimensions are also derived. The instability of the fixed point is discussed. The rigorous results for η and δ with an arbitrary N and for γ and φ in the spherical case N=∞ are obtained.
Critical behaviors in quenched random-spin systems with N-spin component are studied in the limit M → 0 of the non-random MN-component models by means of the renormalization group theory. As the static critical phenomena the stability of the fixed points is investigated and the critical exponents η[~ O(ε3); ε ≡ 4 – d], γ, α, and crossover index [Formula: see text] and the equation of state [~ O(ε)] are obtained. Within the approximation up to the order ε2, even the random-spin systems with N = 2 or 3 are unstable in the three dimensions and the pure systems are stable there.
Tricriticdal behavior, in particular, critical exponents (η, γ, φ, ν, α), of classical N-component spin systems with a short-range or long-range interaction are studied in the dimension d (≪ 3 σ/2; σ: potential range parameter, 1 ≪ σ≤2) by means of the renormalization theory. The obtained expressions for all these critical exponents in the long-range interaction system and for the crossover exponent φ in the short-range one are new. The corrections of these tricritical exponents to the classical ones are shown to be smaller than those in the corresponding systems undergoing the second-order phase transition.
By means of the renormalisation-group approach, static and dynamic critical behaviours are studied for the XY spin system with cubic anisotropy and with quenched random nonmagnetic impurities. The stability of the quenched-random-impurity fixed point in d ( identical to 4- epsilon ) dimensions, the flow diagram of the interactions and the critical exponents are investigated, within the two-loop approximation, in terms of the epsilon 1/2 expansion. It is shown that the transition that appears is a first-order transition.
The renormalization group approach is applied to the study of the short-time critical behavior of the d-dimensional n-component spin systems with cubic anisotropy. First, the system is quenched from high temperature to the critical temperature and then relaxes to equilibrium within model A dynamics. The asymptotic scaling laws and the initial slip exponents ${\ensuremath{\theta}}^{\ensuremath{'}}$ and $\ensuremath{\theta}$ of the order parameter and the response function, respectively, are calculated to the second order in $\ensuremath{\epsilon}=4\ensuremath{-}d.$ Logarithmic corrections to short-time behaviors of both the autocorrelation and the order parameter are found in d=4d=4d=4 dimension.