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Progress of Theoretical Physics

Critical Behavior of Isotropic Systems with Long-Range Interactions

Y. Yamazaki, Masuo Suzuki

Published Jun 1, 1977• 17 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.57.1886

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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.
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.
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.
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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.