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Communications in Theoretical Physics

Tricritical and Critical Exponents in Microcanonical Ensemble of Systems with Long-Range Interactions*

Liangsheng Li

Published Dec 1, 2016• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/66/6/638

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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.
We investigate the thermodynamic quantities of Blume-Emery-Griffiths model with long-range interactions and observe negative specific heat and susceptibility near the first order line, where the temperature, specific heat and susceptibility are discontinuity. When order-disorder phase transition takes place, the critical exponent of specific heat (α=α′=0) and the critical exponent of susceptibility (γ=γ′=1) are consistent with the prediction of mean field theory. However, when coupling constant corresponds to the value of canonical tri-critical point, the critical exponent of specific heat and susceptibility take the values of 0=α≠α′=1 and 1=γ≠γ′=2. The critical exponents violate the scaling homogeneity equalities and α′+β+γ′>2 breaks down the Widom scaling law.
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.
We introduce two variants of the Blume--Emery--Griffiths model with long-range, mean-field-like interactions and quenched disorder. Although the two variants are described by different Hamiltonians, they yield the same free energy. Moreover, both Hamiltonians lead to the same average energy and hence to the same microcanonical entropy. The canonical and microcanonical phase boundaries differ markedly, and so does the multicritical behaviour of the two ensembles. In the microcanonical ensemble, tricritical points appear for any fraction of active spins, whereas in the canonical ensemble they emerge only when this fraction is sufficiently large. At lower fractions, canonical critical endpoints arise instead. We also report the formation of gaps in the entropy as a function of magnetization, another manifestation of ensemble inequivalence.
Abstract Two infinite range Ising models are solved exactly using a microcanonical ensemble in the longrange order parameter. Classical tricritical points exist for both models. The first model involving two‐spin and four‐spin terms is relevant to the compressible Ising model. The other model is the infinite range version of the Blume‐Capel spin‐one Ising model. In both the cases the exact free‐energy functions calculated are examples of exact classical Landau free‐energy functions with tricritical point behaviour.
Critical exponents for a one-dimensional general continuous Ising model with long-range ferromagnetic interactions decaying as 1/r1+σ are calculated using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between −1 and 1. The critical point behavior is investigated. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents; the singularities in the specific heat and susceptibility depend on σ. For σ=1, there are extremely weak finite size effects: the first-order and the second-order cumulants of the order parameter yield ν=2.42(1). The susceptibility exponent γ=2.259(9). Results for σ=0.7 will be shown and discussed.