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 the binary correlation approximation and explicit results for k = 1, we show that for k ≪ m ≪ l, the spectral fluctuations are Poissonian. 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 in the regime 2k m, the embedded ensembles do not possess Wigner–Dyson spectral fluctuation properties.