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Communications in Theoretical Physics

Microcanonical Phase Diagram of the BEG and Ising Models

Liangsheng Li, Ning Zheng, Qingfan Shi

Published Sep 1, 2012• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/58/3/22

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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.
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.
A linked-cluster expansion technique for spin systems has been used to obtain the free energy, the magnetization, and the susceptibility series for three Ising systems: (1) The Ising model; (2) The Blume-Capel model; and (3) The Blume-Emery-Griffiths (BEG) model. Each Hamiltonian is first divided into a single-ion potential and a term describing the interaction of the spin fluctuations. Only the latter is treated as perturbation in the series expansion. In the parametric phase our series reduces to the exact high-temperature series. However, in the ordered phase the same series can be used to study the low temperature behavior, such as the variation of the magnetization with temperature. The analysis of the eighth-order series for systems (1) and (2) show results in excellent agreement with the previous ones obtained from the high-temperature and low-temperature series. The series analysis for the BEG model shows quite different behavior from that predicted by the mean-field approximation.
A phase diagram is a graph in parameter space showing the phase boundaries of a many-particle system. Commonly, the control parameters are chosen to be those of the (generalized) canonical ensemble, such as temperature and magnetic field. However, depending on the physical situation of interest, the (generalized) microcanonical ensemble may be more appropriate, with the corresponding control parameters being energy and magnetization. We show that the phase diagram on this parameter space looks remarkably different from the canonical one. The general features of such a microcanonical phase diagram are investigated by studying two models of ferromagnets with short-range interactions. The physical consequences of the findings are discussed, including possible applications to nuclear fragmentation, adatoms on surfaces, and cold atoms in optical lattices.
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 study the phase diagram of the 3-matrix model. This model corresponds to the Blume-Emery-Griffiths (BEG) model on a random surface. The BEG model has parameters J and K which are the spin and the lattice-gas coupling respectively. If K = 0, it is the usual tri-critical Ising model. If K/J = 3, this model becomes the 3-state Potts model. These models are solved in the context of the Coulomb gas representation. The intermediate region K/J = 1 is equivalent to the 3-matrix chain model and we can solve it exactly using the orthogonal polynomial method. We find the 3rd order critical line on which the spin ordering phase transition occurs. This line ends at the 4th order critical point which corresponds to the tri-critical point of the BEG model. The phase diagram is quite similar to that of the BEG model on a regular lattice.
A long-range interacting Fermi chain placed in the uniform and the staggered magnetic field is studied via the micro-canonical approach. The relation between the entropy and the energy of the system is obtained by counting the number of microscopic states. We find that this system is non-ergodic and can exhibit first-order phase transition, second-order phase transition, or both. The microcanonical ensemble predicts negative specific heat regions and temperature jumps. Moreover, the global phase diagram of the system is constructed.