  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.
