Zeros on the temperature axis of spin correlations in a random bond Ising chain

Abstract

The authors study the correlation between two spins diametrically opposite in a random Ising loop of 2r spins with random nearest-neighbour couplings. As a function of the inverse temperature beta , this correlation undergoes random sign changes with a density rho r( beta ). They determine this density for an arbitrary distribution P(K) of the coupling constants for (i) r to infinity and (ii) the scaling limit beta to infinity , r to infinity at r/ beta fixed. For r to infinity , the total number of zeros on the beta -axis grows asymptotically as 1/2 1/21 zeta (3)-1/4 pi 21/2 log r. As an application, the density of zeros is calculated for the spin-spin correlation in a double infinite Ising chain with random bonds, having, with a probability p, an infinite transverse coupling between each pair of corresponding sites. A connection is made with predictions from spin glass theory.