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We study the nature of the broken ergodicity in the low temperature phase ofIsing spin glass systems, using as a diagnostic tool the spectrum ofeigenvalues of the spin-spin correlation function. We show that multipleextensive eigenvalues of the correlation matrix $C_{ij}\equiv< S_i S_j>$ occurif and only if there is replica symmetry breaking. We support our argumentswith Exchange Monte-Carlo results for the infinite-range problem. Here we findmultiple extensive eigenvalues in the RSB phase for $N \agt 200$, but only asingle 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 derivedfrom the droplet model. Because of the small system sizes we cannot exclude thepossibility of replica symmetry breaking with finite size corrections in thisregime.Comment: 14 pages, 7 figures, submitted to PR

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