Extensive eigenvalues in spin-spin correlations: a tool for counting pure states in Ising spin glasses

We study the nature of the broken ergodicity in the low temperature phase of Ising spin glass systems, using as a diagnostic tool the spectrum of eigenvalues of the spin-spin correlation function. We show that multiple extensive eigenvalues of the correlation matrix $C_{ij}\equiv< S_i S_j>$ occur if and only if there is replica symmetry breaking. We support our arguments with Exchange Monte-Carlo results for the infinite-range problem. Here we find multiple extensive eigenvalues in the RSB phase for $N \agt 200$, but only a single extensive eigenvalue for phases with long-range order but no RSB. Numerical results for the short range model in four spatial dimensions, for $N\le 1296$, are consistent with the presence of a single extensive eigenvalue, with the subdominant eigenvalue behaving in agreement with expectations derived from the droplet model. Because of the small system sizes we cannot exclude the possibility of replica symmetry breaking with finite size corrections in this regime.