The nature of ergodicity breaking in Ising spin glasses as revealed by correlation function spectral properties

In this work we address the nature of broken ergodicity in the low temperature phase of Ising spin glasses by examining spectral properties of spin correlation functions $C_{ij}\equiv< S_i S_j>$. We argue that more than one extensive (i.e., O(N)) eigenvalue in this matrix signals replica symmetry breaking. Monte-Carlo simulations of the infinite-range Ising spin-glass model, above and below the Almeida-Thouless line, support this conclusion. Exchange Monte-Carlo simulations for the short-range model in four dimensions find a single extensive eigenvalue and a large subdominant eigenvalue consistent with droplet model expectations.