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Physical review. B, Condensed matter

Griffiths singularities in random magnets: Results for a soluble model

Alan J. Bray, Deng Huifang

Published Oct 1, 1989• 53 citations
Publisher

Abstract

A soluble, but nontrivial, model of a dilute Ising ferromagnet is studied, with infinite-range interactions but finite average connectivity c. The density of (Yang-Lee) zeros of the partition function in the complex z=exp(-2H) plane (where H is the external magnetic field in units of the temperature) is calculated explicitly in the high-temperature phase for large but finite c and small \ensuremath{\Vert}H\ensuremath{\Vert}. The density of zeros on the unit circle H=i\ensuremath{\theta} has the form \ensuremath{\rho}(\ensuremath{\theta})\ensuremath{\sim}exp{-[cf(K)/\ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert}]ln/B (1/\ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert})} for \ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert}\ensuremath{\rightarrow}0. The function f(K) (K=J/T) vanishes at the critical coupling ${K}_{C}$(c). Heuristic arguments are given for the form of \ensuremath{\rho}(\ensuremath{\theta}) expected for systems with short-range interactions.

DOI
10.1103/physrevb.40.6980

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A soluble, but nontrivial, model of a dilute Ising ferromagnet is studied, with infinite-range interactions but finite average connectivity c. The density of (Yang-Lee) zeros of the partition function in the complex z=exp(-2H) plane (where H is the external magnetic field in units of the temperature) is calculated explicitly in the high-temperature phase for large but finite c and small \ensuremath{\Vert}H\ensuremath{\Vert}. The density of zeros on the unit circle H=i\ensuremath{\theta} has the form \ensuremath{\rho}(\ensuremath{\theta})\ensuremath{\sim}exp{-[cf(K)/\ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert}]ln/B (1/\ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert})} for \ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert}\ensuremath{\rightarrow}0. The function f(K) (K=J/T) vanishes at the critical coupling KC{K}_{C}KC​(c). Heuristic arguments are given for the form of \ensuremath{\rho}(\ensuremath{\theta}) expected for systems with short-range interactions.
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.
We apply renormalization-group methods to study the density of Yang-Lee zeros at the edges of the gap in Ising systems with correlated random-Tc{T}_{c}Tc​ impurities. The impurity correlations are assumed to fall off at large distances as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}\ensuremath{\theta})}$ (\ensuremath{\theta}>0). Both short- and long-range spin interactions decaying as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d+\ensuremath{\sigma})}$ (0\ensuremath{\sigma}2) are considered. We find that the edge singularities are described by a new fixed point in the underlying ${\ensuremath{\varphi}}^{3}$-field model with imaginary coupling and imaginary correlated random fields. We obtain the edge exponents to leading order in the \ensuremath{\epsilon}\ifmmode \tilde{}\else \~{}\fi{} expansion, where \ensuremath{\epsilon}\ifmmode \tilde{}\else \~{}\fi{}=dc{d}_{c}dc​-d with dc{d}_{c}dc​=8+\ensuremath{\theta} for short- and dc{d}_{c}dc​=4\ensuremath{\sigma}+\ensuremath{\theta} for long-range interactions which do not obey any apparent dimensional reduction rule.
Arguments are given that, for random spin systems, the density of states \ensuremath{\rho}(\ensuremath{\mu}) of the inverse of the susceptibility matrix vanishes as \ensuremath{\rho}(\ensuremath{\mu})\ensuremath{\sim}exp(-A/\ensuremath{\mu}), for \ensuremath{\mu}\ensuremath{\rightarrow}0, throughout the ``Griffiths phase.'' The amplitude A vanishes at the onset of magnetic long-range order, and diverges at the transition between ``Griffiths'' and ``paramagnetic'' phases. For an O(m) spin system, with m\ensuremath{\rightarrow}\ensuremath{\infty}, the spin autocorrelation function C(t) is found to have the ``stretched-exponential'' form, lnC(t)\ensuremath{\sim}-(At)1/2{)}^{1/2})1/2, in the Griffiths phase.
We study the diluted Ising ferromagnet on the Bethe lattice as a case study for the application of the cavity method to problems with Griffiths-McCoy singularities. Specifically, we are able to make much progress at infinite coupling where we compute, from the cavity method, the density of Lee-Yang zeros in the paramagnetic Griffiths region as well as the properties of the phase transition to the ferromagnet. This phase transition is itself of a Griffiths-McCoy character albeit with a power law distribution of cluster sizes.
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.
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).