M ay 2 00 1 Spectral Properties of the k – Body Embedded Gaussian Ensembles of Random Matrices

We consider m spinless Fermions in l > m degenerate single– particle levels interacting via a k–body random interaction with Gaus-sian probability distribution and k ≤ m in the limit l → ∞ (the embedded k–body random ensembles). We address the cases of orthogonal and unitary symmetry. We derive a novel eigenvalue expansion for the second moment of the Hilbert–space matrix elements of these ensembles. Using properties of the expansion and the supersymmetry technique, we show that for 2k > m, the average spectrum has the shape of a semicircle, and the spectral fluctuations are of Wigner– Dyson type. Using a generalization of the binary correlation approximation , we show that for k ≪ m ≪ l, the spectral fluctuations are Poissonian. This is consistent with the case k = 1 which can be solved explicitly. We construct limiting ensembles which are either fully in-tegrable or fully chaotic and show that the k–body random ensembles lie between these two extremes. Combining all these results we find that the spectral correlations for the embedded ensembles gradually change from Wigner–Dyson for 2k > m to Poissonian for k ≪ m ≪ l.