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arXiv:2208.12965 · Physical review. E

Ensemble dependence of the critical behavior of a system with long-range interaction and quenched randomness

Nir Schreiber, Reuven Cohen C, Simi Haber

Published Aug 14, 2023• 0 citations
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Abstract

We propose a hybrid model governed by the Blume-Emery-Griffiths (BEG) Hamiltonian with a mean-field-like interaction, where the spins are randomly quenched such that some of them are "pure" Ising and the others admit the BEG set of states. It is found, by varying the concentration of the Ising spins, that the model displays different phase portraits in concentration-temperature parameter space, within the canonical and the microcanonical ensembles. Phenomenological indications that these portraits are rich and rather unusual are provided.

DOI
10.1103/physreve.108.024124

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We propose a hybrid model governed by the Blume-Emery-Griffiths (BEG) Hamiltonian with a mean-field-like interaction, where the spins are randomly quenched such that some of them are "pure" Ising and the others admit the BEG set of states. It is found, by varying the concentration of the Ising spins, that the model displays different phase portraits in concentration-temperature parameter space, within the canonical and the microcanonical ensembles. Phenomenological indications that these portraits are rich and rather unusual are provided.
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.
The mixed spin-2 and spin-3/2 Blume-Emery-Griffiths (BEG) Ising ferrimagnetic system is studied by the Bethe lattice approach. The ground-state phase diagram is constructed. The influence of the crystal-field and the biquadratic interactions among neighboring spins on the thermal behaviors of the system is singled out. The system displays very rich critical behaviors with the existence of tricritical points. Compensation points where the global magnetization of the system vanishes have been detected for appropriate values of the system parameters.
The effect of the random quantum transverse field Ω on the tricritical behavior of the spin-1 Blume–Emery–Griffiths (BEG) model is studied using an effective field theory. It is found that the tricritical behavior depends on both the biquadratic interaction K, single-ion anisotropy Δ and the concentration p of the disorder of Ω. Indeed, there exists a special value p1 of the probability p below which the tricritical behavior disappears. In addition, at sufficiently low temperatures, the system exhibits long-range order accompanied by the tricritical behavior below a special value p2 of the probability p.
An extensive study of the mean-field static solution of the random Blume-Emery-Griffiths-Capel model, an Ising-spin lattice gas with quenched random magnetic interaction, is performed. The model exhibits a paramagnetic phase, described by a stable replica-symmetric solution. When the temperature is decreased or the density increases, the system undergoes a phase transition to a full-replica-symmetry-breaking spin-glass phase. The nature of the transition can be either of the second order (such as in the Sherrington-Kirkpatrick model) or, at temperatures below a given critical value, of the first order in the Ehrenfest sense, with a discontinuous jump of the order parameter and accompanied by a latent heat. In this last case, coexistence of phases takes place. The thermodynamics is worked out in the full-replica-symmetry-breaking scheme, and the related Parisi equations are solved using a pseudospectral method down to zero temperature.
The density of states of long-range Blume—Emery—Griffiths (BEG) and short-range Ising models are obtained by using Wang—Landau sampling with adaptive windows in energy and magnetization space. With accurate density of states, we are able to calculate the microcanonical specific heat of fixed magnetization introduced by Kastner et al. in the regions of positive and negative temperature. The microcanonical phase diagram of the Ising model shows a continuous phase transition at a negative temperature in energy and magnetization plane. However the phase diagram of the long-range model constructed by peaks of the microcanonical specific heat looks obviously different from the Ising chart.
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.