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International Journal of Modern Physics C

SUSCEPTIBILITY SINGULARITIES AT FIRST-ORDER PHASE TRANSITIONS

Douglas B. Abraham, Paul J. Upton

Published Oct 1, 1992• 2 citations
Publisher

Abstract

Problems associated with analyticity of thermodynamic functions close to first-order phase transitions are briefly reviewed. The bubble model for correlation functions is then applied to planar Ising-like models at subcritical temperatures (T<Tc) with a bulk magnetic field h. The fluctuation sum is used to calculate the susceptibility χ(h) from the bubble correlation function. We show that χ(h), calculated this way, must contain an essential singularity at h=0 i.e. at the first-order phase boundary. This has important implications to metastability, where we demonstrate that if the ensemble is restricted such that the magnetization stays positive when h goes negative, χ(h) has an infinite number of poles along the negative real axis with a limit-point at h=0. For an unrestricted ensemble, a Yang-Lee circle theorem is derived.

DOI
10.1142/s0129183192000701

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Problems associated with analyticity of thermodynamic functions close to first-order phase transitions are briefly reviewed. The bubble model for correlation functions is then applied to planar Ising-like models at subcritical temperatures (T<Tc) with a bulk magnetic field h. The fluctuation sum is used to calculate the susceptibility χ(h) from the bubble correlation function. We show that χ(h), calculated this way, must contain an essential singularity at h=0 i.e. at the first-order phase boundary. This has important implications to metastability, where we demonstrate that if the ensemble is restricted such that the magnetization stays positive when h goes negative, χ(h) has an infinite number of poles along the negative real axis with a limit-point at h=0. For an unrestricted ensemble, a Yang-Lee circle theorem is derived.
The bubble picture for pair-spin correlation functions is applied to two-dimensional Ising-like models, at subcritical temperatures, in an external field h, from which we show, via the fluctuation sum, that the susceptibility \ensuremath{\chi}(h) has an essential singularity at h=0. This is studied further using a solid-on-solid bubble model for (a) a restricted ensemble corresponding to metastability, where h=0 is found to be the limit point of an infinite number of poles of \ensuremath{\chi} along the negative real axis, and (b) an unrestricted ensemble, where a Yang-Lee circle theorem is found.
Abstract We address a number of outstanding questions associated with the analytic properties of the universal equation of state of the ϕ 4 theory, which describes the critical behavior of the Ising model and ubiquitous critical points of the liquid–gas type. We focus on the relation between spinodal points that limit the domain of metastability for temperatures below the critical temperature, i.e. T < T c , and Lee-Yang edge singularities that restrict the domain of analyticity around the point of zero magnetic field H for T > T c . The extended analyticity conjecture (due to Fonseca and Zamolodchikov) posits that, for T < T c , the Lee-Yang edge singularities are the closest singularities to the real H axis. This has interesting implications, in particular, that the spinodal singularities must lie off the real H axis for d < 4 , in contrast to the commonly known result of the mean-field approximation. We find that the parametric representation of the Ising equation of state obtained in the ε = 4 − d expansion, as well as the equation of state of the O ( N ) -symmetric ϕ 4 theory at large N , are both nontrivially consistent with the conjecture. We analyze the reason for the difficulty of addressing this issue using the ε expansion. It is related to the long-standing paradox associated with the fact that the vicinity of the Lee-Yang edge singularity is described by Fisher’s ϕ 3 theory, which remains nonperturbative even for
By an exact calculation on the planar Ising model we show that for all subcritical temperatures, the bubble model predicts the correct average distribution of matter contributing to the pair-spin correlation function in the two-particle sector. This considerably substantiates the bubble model which is generalized to higher dimensions with an external field h and then used to show that the free energy has an essential singularity in h of Andreev-Fisher form at all points along the phase boundary. We argue that such a singularity would also occur in the Baxter model.
There exists the famous circle theorem on the Yang-Lee zeros of the ferromagnetic Ising model. However, the Yang-Lee zeros of the antiferromagnetic Ising model are much less well understood than those of the ferromagnetic model. The precise distribution of the Yang-Lee zeros of the antiferromagnetic Ising model only with nearest-neighbor interaction J on LxL square lattices is determined as a function of temperature a=e(2betaJ) (J<0), and its relation to the phase transitions is investigated. In the thermodynamic limit (L-->infinity), the distribution of the Yang-Lee zeros of the antiferromagnetic Ising model cuts the positive real axis in the complex x=e(-2betaH) plane, resulting in the critical magnetic field +/-H(c)(a), where H(c)>0 below the critical temperature a(c)=square root of 2-1. The results suggest that the value of the scaling exponent y(h) is 1 along the critical line for a<a(c).
Expressions for general correlation functions in Ising ferromagnets are derived in terms of the Lee-Yang theorem. By the use of these expressions, the asymptotic forms of correlation functions in the presence of magnetic field are analyzed, assuming those in the absence of the external field. In particular, the critical indices of spin-spin, energy-density-energy-density, energy-density-spin correlation functions are related with the critical indices of thermal susceptiblity χT0, specific heat, susceptibility χH0, and magnetization. These results agree in form with those predicted by the scaling law approach.
The Ising ferromagnet, consisting of magnetic spins, is the simplest system showing phase transitions and critical phenomena at finite temperatures. The Ising ferromagnet has played a central role in our understanding of phase transitions and critical phenomena. Also, the Ising ferromagnet explains the gas-liquid phase transitions accurately. In particular, the Ising ferromagnet in a nonzero magnetic field has been one of the most intriguing and outstanding unsolved problems. We study analytically the partition function zeros in the complex magnetic-field plane and the Yang-Lee edge singularity of the infinite-range Ising ferromagnet in an external magnetic field. In addition, we compare the Yang-Lee edge singularity of the infinite-range Ising ferromagnet with that of the square-lattice Ising ferromagnet in an external magnetic field.