The complete thermodynamics of a model hamiltonian with general-spin infinite range Ising 2-point, 4-point,... interaction terms is found exactly in two different statistical mechanical ensembles which prove to be essentially equivalent. The coefficients of the above terms may be chosen to give critical behaviour which is largely consistent with the idea of scaling and which has delta equal to any odd positive integer and gamma =1; alternatively a first order phase transition may be obtained, a finite spontaneous magnetization appearing as the temperature is decreased through that of the transition. The Heisenberg case is also considered and found to be closely similar to the Ising case.
The complete thermodynamics of a model hamiltonian with general-spin infinite range Ising 2-point, 4-point,... interaction terms is found exactly in two different statistical mechanical ensembles which prove to be essentially equivalent. The coefficients of the above terms may be chosen to give critical behaviour which is largely consistent with the idea of scaling and which has delta equal to any odd positive integer and gamma =1; alternatively a first order phase transition may be obtained, a finite spontaneous magnetization appearing as the temperature is decreased through that of the transition. The Heisenberg case is also considered and found to be closely similar to the Ising case.
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.
In Chapter 3, we examined two examples of phase transitions, the paramagnetic–ferromagnetic transition (Section 3.1.4) and the liquid–vapour transition (Section 3.5.4). In the latter case, thermodynamic functions such as entropy or specific volume are discontinuous at the transition. When, for example, one varies the temperature T at constant pressure, the transition takes place at a temperature T c where the Gibbs potentials of the two phases are equal, but the two phases coexist at T c and are either stable or metastable in the vicinity of T c . Each of the phases carries with it its own entropy, specific volume, etc., which are in general different at T = T c , hence the discontinuities. Such a phase transition is called a first order phase transition . The picture is quite different in the paramagnetic–ferromagnetic phase transition: the thermodynamic quantities are continuous at the transition, the transition is not linked to the crossing of two thermodynamic potentials and one never observes metastability. Such a transition is called a second order, or continuous, phase transition . The transition temperature is called the critical temperature and is denoted by T c . The new and remarkable feature which we shall encounter in continuous phase transitions is the existence of cooperative phenomena. To be specific, think of a spin system, for example an Ising or a Heisenberg model, where the interactions between spins are limited to nearest neighbours. More generally we shall consider short range interactions , which decrease rapidly as a function of the distance between spins, and we exclude from our study all long range interactions.
The method of developing exact power-series expansions for the partition function ZN and related thermodynamic functions for the Ising model valid below the critical point is generalized to include exchange interactions between first-, second-, and third-neighbor pairs. Expansions of the spontaneous magnetization M0(T) and zero field susceptibility χ0(T) are derived through to sixth order of perturbation for the s.q. lattice, and through to fifth order of perturbation for the Δ′r, b.c.c., s.c., and f.c.c. lattices, when interactions J1SizSjz and J2SkzSlz are present between first- and second-neighbor spins, respectively (second-neighbor model). These expansions have also been obtained for the case where interactions of equal magnitude (J1 = J2 = J3) are present between first-, second-, and third-neighbor pairs (third-equivalent-neighbor model); here expansions through to fifth order of perturbation are obtained for the s.q., Δ′r, b.c.c., and s.c. lattices and through to fourth order for the f.c.c. lattice. The Padé approximant·procedure is employed to discuss the effects of an extended but finite range of interaction on the behavior of M0(T) and χ0(T) for T→Tc− as characterized by the critical exponents β and γ′, respectively. For the second-equivalent-neighbor model lattices, it is found that 0.122 ≤ β ≤ 0.134 in two dimensions, and that 0.308 ≤ β ≤ 0.328 in three dimensions; from which it is concluded that β remains unchanged from its value in the nearest-neighbor model. The corresponding limits for γ′ in three dimensions are 1.18 ≤ γ′ ≤ 1.28; from this and the results for the b.c.c. lattice in particular, it is concluded that γ′ is probably 54 and hence the transition in χ0 is symmetrical about Tc (γ′ = γ). A repetition of this analysis for the third-equivalent-neighbor model three-dimensional lattices shows a marked shift in the estimated range of β and γ′; the results are 0.345 ≤ β ≤ 0.365, and 1.01 ≤ γ′ ≤ 1.14. In each of the above cases, the corresponding high-temperature (T > Tc) expansions of χ0(T) obtained previously have been analyzed to yield estimates of the critical exponent γ. The over-all results and in particular the estimates of γ for the s.q. and b.c.c. lattices suggest that this index is unaffected by extending the range of interaction, and that if γ is a rational fraction then it is the same fraction for the n.n. model and second- and third-equivalent neighbor models. Finally the high-temperature expansions of ZN in zero field, and of χ0(T) for the second-neighbor model are used to examine the dependence of the critical temperature Tc, the critical energy (E∞ − Ec)/kTc, and the critical entropy (S∞ − Sc)/k on the relative strengths of J1 and J2 for values of J2/J1 in the range 0 to 1. It is found that the variation of the critical point is well represented by Tc(α)=Tc(0)[1+m1α],where α = J2/J1 and lies in the range 0 ≤ α ≤ 1; and Tc(0) is the critical temperature of the nearest-neighbor model. The values of m1 are 0.61, 2.47, 0.84, 1.45, and 1.35 for the f.c.c., s.c., b.c.c., s.q., and Δ′r lattices, respectively. All these calculations are compared with the corresponding results for the Heisenberg model.
The problem of finding a microscopic theory of phase transitions across a critical point is a central unsolved problem in theoretical physics. We find a general solution to that problem and present it here for the cases of Bose–Einstein condensation (BEC) in an interacting gas and ferromagnetism in a lattice of spins, interacting via a Heisenberg or Ising Hamiltonian. For BEC, we present the exact, valid for the entire critical region, equations for the Green's functions and order parameter, that is a critical-region extension of the Beliaev–Popov and Gross–Pitaevskii equations. For the magnetic phase transition, we find an exact theory in terms of constrained bosons in a lattice and obtain similar equations for the Green's functions and order parameter. In particular, we outline an exact solution for the three-dimensional Ising model.
Phase transitions in Ising models with tree surfaces are studied from various points of view, including a phenomenological Landau theory, high-temperature series expansions, and a scaling theory for thermodynamic quantities and correlation functions. In the presence of a surface a number of new critical exponents must be defined. These arise because of the existence of "surface" terms in the thermodynamic functions, and because of the anisotropy of space and lack of translational symmetry introduced by the surface. The need for these new critical exponents already appears in the phenomenological theory, which is discussed in detail and related to the microscopic mean-field approximation. The essential new parameter appearing in this theory is an extrapolation length $\ensuremath{\lambda}$ which enters the boundary condition on the magnetization at the surface. For magnetic systems this length is of the order of the interaction range, in contrast to superconductors, where it is usually much larger. In order to go beyond the mean-field theory, high-temperature series expansions are carried out for the Ising half-space, to tenth order in two dimensions and to eighth order in three dimensions. A scaling theory is developed both for thermodynamic functions and for spin correlations near the surface, and relations are found among the exponents of the half-space. Both the scaling theory and the numerical calculations are compared with the exact solution of the Ising half-plane (two dimensions) by McCoy and Wu, and agreement is found wherever the theory is applicable. In analogy to the bulk situation, the scaling theory is found to agree with mean-field theory in four dimensions. The prediction of the present work which is most easily accessible to experiment is the temperature dependence of the magnetization at the surface, with critical exponent estimated to be ${\ensuremath{\beta}}_{1}=\frac{2}{3}$. The mean-field result, ${\ensuremath{\beta}}_{1}=1$, seems to agree more closely with presently available experiment, and more work is needed to clarify the situation.