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

In this Letter we address the nature of broken ergodicity in the low temper ature phase of Ising spin glasses by examining spectral properties of spin correlation functions C-ij drop (SiSj). We argue that more than one extensi ve [i.e., O(N)] eigenvalue in this matrix signals replica symmetry breaking . Monte Carlo simulations of the infinite-range Ising spin-glass model, abo ve and below the Almeida-Thouless line, support this conclusion. Exchange M onte Carlo simulations for the short-range model in four dimensions find a single extensive eigenvalue and a large subdominant eigenvalue consistent w ith droplet model expectations.