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 Gaussian probability distribution and k less than or equal to m in the limit l --> infinity (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 much less than m much less than 1, 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 integrable 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 much less than m much less than 1. (C) 2001 Academic Press.