Correlation functions of the Ising spin glass on the Bethe lattice

Abstract

We analyze the properties of the ordered phase of the Ising spin glass on the Bethe lattice. In particular, we obtain the length dependence of the probability distribution of droplet excitation energies and calculate the truncated two-spin correlation function approximately in the limit of low temperatures. For comparative purposes, we study both unfrustrated random-exchange magnets and spin glasses with continuous, symmetrical exchange distributions. We find that, as T\ensuremath{\rightarrow}0, the correlation length associated with the averaged correlation function approaches zero for unfrustrated systems, whereas for spin glasses this correlation length approaches a nonzero constant. The correlation length associated with the typical correlation function approaches zero as T\ensuremath{\rightarrow}0 for both spin glasses and unfrustrated random-exchange magnets. We discuss the divergence of various susceptibilities near T=0 and the existence of infinite droplet excitations of finite energy. We also discuss the possible relationship of our results to finite-dimensional spin-glass models.