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Progress of Theoretical Physics

Critical Behavior in Tricritical Phase Transition. I: Critical Exponents ( , , , , )

Y. Yamazaki, Masuo Suzuki

Published Aug 1, 1977• 3 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.58.516

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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.
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.
Abstract We explore the tricritical points and the critical lines of both Blume–Emery–Grifnths and Ising model within long-range interactions in the microcanonical ensemble. For K = K MTP , the tricritical exponents take the values β = 1/4, 1 = γ − ≠ γ + = 1/2 and 0 = α − ≠ α + = −1/2, which disagree with classical (mean held) values. When K > K MTP , the phase transition becomes second order and the critical exponents have classical values except close to the canonical tricritical parameters (K CTP ), where the values of the critical expoents become β = 1/2, 1 = γ − ≠ γ + = 2 and 0 = α − ≠ α + = 1.
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.
Critical exponents are functions of ddd, the dimension, and nnn, the number of components of the order parameter involved in the second-order phase transition. They can be expanded in power series of 1n\frac{1}{n}n1​ when nnn is large. We present details of calculating by perturbation theory the critical exponents above Tc{T}_{c}Tc​ for arbitrary ddd and to O(1n)O(\frac{1}{n})O(n1​) for short-range interacting systems and also for long-range interacting systems.