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Progress of Theoretical Physics

Order of the Phase Transition Related to the Interaction and the Degeneracy

H. Nakano

Published Jun 1, 1975• 7 citations
Publisher

Abstract

It is investigated how the order of phase transition in a system of many identical interacting subsystems depends on the degeneracies of the states of each subsystem and on the interaction between subsystems; in particular, on the symmetry and uniformity of the degeneracy and on the symmetry of the interaction. By denoting the state of the i-th subsystem by a spin-like variable σi (i = 1,2,…,N), the degeneracy is expressed as D(σi) as a function of σi and the interaction between the i-th and j-th subsystems as V(σi,σj). (In the text use is made of such a normalized value as ΣσiD(σi) = 1). Then the symmetry means the equalities D(σi) = D(-σi) and V(σi,σj) = V(-σi,-σj) and the uniformity of degeneracy means the independency of D(σi) upon σi. The system of magnetic spins with exchange interactions belongs to the uniform symmetric category, which is used as a reference standard system in the present paper. The thermostatistical average of σi which is denoted by σ represents the order parameter in the system. Dependence of σ upon temperature and some applied field are investigated to see the features of the phase transition. In the non-uniform symmetric case it is found that the phase transition of first order can occur for an adequate form of D(σi), and that it is necessarily of second order in the uniform symmetric case. The induced transition by the applied field is also possible in the former case. In the asymmetric case the phase transition of a peculiar type as well as the induced transition may occur. Under some respective conditions on the degeneracy and interaction parameters, transitions of first and second orders can occur. Analytic properties of the partition functions are also discussed. An exact problem is proposed for the simplest model which is expected to exhibit the first-order phase transition.

DOI
10.1143/ptp.53.1566

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It is investigated how the order of phase transition in a system of many identical interacting subsystems depends on the degeneracies of the states of each subsystem and on the interaction between subsystems; in particular, on the symmetry and uniformity of the degeneracy and on the symmetry of the interaction. By denoting the state of the i-th subsystem by a spin-like variable σi (i = 1,2,…,N), the degeneracy is expressed as D(σi) as a function of σi and the interaction between the i-th and j-th subsystems as V(σi,σj). (In the text use is made of such a normalized value as ΣσiD(σi) = 1). Then the symmetry means the equalities D(σi) = D(-σi) and V(σi,σj) = V(-σi,-σj) and the uniformity of degeneracy means the independency of D(σi) upon σi. The system of magnetic spins with exchange interactions belongs to the uniform symmetric category, which is used as a reference standard system in the present paper. The thermostatistical average of σi which is denoted by σ represents the order parameter in the system. Dependence of σ upon temperature and some applied field are investigated to see the features of the phase transition. In the non-uniform symmetric case it is found that the phase transition of first order can occur for an adequate form of D(σi), and that it is necessarily of second order in the uniform symmetric case. The induced transition by the applied field is also possible in the former case. In the asymmetric case the phase transition of a peculiar type as well as the induced transition may occur. Under some respective conditions on the degeneracy and interaction parameters, transitions of first and second orders can occur. Analytic properties of the partition functions are also discussed. An exact problem is proposed for the simplest model which is expected to exhibit the first-order phase transition.
Phase transitions for an open system consisting of an ensemble of interacting quantum subsystems with discrete spectrum are studied in the mean-field approximation. In the considered model, the change of an internal symmetry of a thermodynamic system upon the second-order phase transition is due to changing symmetry of distribution of charge/spin density inside each quantum subsystem. The latter can be caused by either splitting of one of lowest degenerated energy level or closing a gap between the levels and appearance of avoided crossing. The effect of external parameters (pressure, field, composition, etc.) results in direct change of internal control parameters: level spacing and/or the strength of interaction between adjacent quantum subsystems. Considering a simplest case of the two-level quantum subsystems, expressions for the free energy as a function of the internal control parameters were obtained in analytical form. The behavior of the heat capacity and susceptibility for different regions of the low-temperature phase diagram including the area of quantum fluctuations was determined.
We have considered the paramagnetic—ferromagnetic transition for the Ising system with ``infinite range'' interactions (i.e., each spin interacts equally with all other spins) and in the presence of a single-ion anisotropy term: H=−2jΣ lim i>i SizSjz+ΔΣ lim i Siz2−gμHΣ lim iSiz. Kittel and Shore1 have shown that this system (in the absence of the term in Δ) is exactly soluble, and leads rigourously, in the limit of an infinite number of particles, to the equations of the molecular-field approximation, and to a second-order phase transition. The single-ion term can cause the transition to become first order2,3 in that there is a discontinuity in the magnetization as a function of temperature. We have solved for the relevant thermodynamic parameters as a function of temperature for the case of ions with spin one. The development of the first-order transition as a function of the number of particles will be discussed. This work will be submitted for publication in the Physical Review.
The distribution of zeros of the partition function for one-dimensional lattice system of generalized Ising spins with magnitude unity is studied, where it is assumed that the component spins interact with the nearest neighbours through a ferromagnetic coupling and each spin state has a certain degeneracy. The spin states are denoted by variable σ which can take either of three values, plus and minus unities and zero with the respective degeneracies, D(1), D(0) and D(-1), where it is assumed D(1) = D(-1). zeros of the partition function Z(ζ) of the system as exposed to an external magnetic field are studied on the ζ-plane, where ζ=exp(H) indicates the exponential function of the external field divided by temperature H and it is regarded as a complex number. The distribution of zeros is classified into three typical patterns, corresponding to the respective three regions on the plane of temperature(T) versus degneracy ratio (ρ=D(0)/D(1)): Zeros never lie on the unit circle in the region of larger ρ, all on the unit circle in the region of smaller ρ and only a part on the unit circle in the remaining region. On the basic of this result, it is expected in the case of two- or three-dimensional system that phase trasitions of first and second orders occur in tha cases of ρ larger and smaller than acritical calue ρc, respectively.
This chapter explores the basic notions of phase transitions, and the manner in which non-analytic thermodynamic behaviour may occur in principle. It presents some concrete calculations of phase transition behaviour in magnetic systems. The chapter is concerned with the transfer matrix method, first introduced by Hendrik Kramers, which reduces the problem of calculating the partition function to the problem of finding the eigenvalues of a certain matrix. It describes the low temperature expansion and Weiss’ mean field theory for magnets. Mean field theory is the simplest treatment of an interacting statistical mechanical system, apart from the approximation of ignoring the interactions all together. The chapter provides a simple, non-systematic derivation of mean field theory for the Ising model. This mean field theory can be easily improved by applying it to clusters of spins rather than individual spins.
Averaging over interaction fields and spin correlations in the Ising modelIn the theory of phase transitions in magnetic systems, there are a small number of exact results.One of them is the well-known Onsager solution for the Ising model on a square lattice.The problem of phase transitions in magnets is a special case of the problem of many bodies.There are many approaches and methods in this area, including the method of averaging over exchange fields.The application of the distribution function over interaction fields to the study of the properties of a system of many interacting particles has been used for a long time.In previous works of the authors, the method of averaging over exchange fields was applied to the analysis of the magnetic properties of pure and diluted magnets.In this paper, we formulate and prove the relations on which the method of averaging over exchange fields can be based on clusters of several spins.It turns out that based only on the possibility of representing thermodynamic means as means of the distribution function of the fields, some nontrivial exact relations can be obtained.The paper considers the Ising model on simple lattices with coordination numbers of three and four.Exact expressions are found that connect the average products of three spins with spontaneous magnetization in the system.Inequalities are obtained that limit the limiting value of the Curie temperature and the maximum value of spontaneous magnetization at a given temperature.
The problem of N interacting spins on a lattice is equivalent to one of N clusters linked in a specific manner. The energy of any configuration of spins can be expressed in terms of the energy levels of this cluster. A new expression is obtained for the probability of occurrence of any configuration. A closed form expression is obtained for the partition function per spin in terms of the energy levels of this cluster with the degeneracies being a function of temperature. On physical grounds it is suggested to be the form of the exact partition function per spin. The partition functions of Ising-like systems all have a common form. For the 3D Ising model seven functions need to be determined to describe the partition function completely. The key to understanding second order phase transitions and critical phenomena lies in the temperature dependence of various degeneracies. It is necessary to develop new techniques to determine the partition function that account for this temperature dependence, as it would represent the underlying physics correctly.