Spectral Properties of the k-Body Embedded Gaussian Ensembles of Random Matrices

Abstract

We consider $m$ spinless Fermions in $l>m$ degenerate single-particle levels interacting via a $k$-body random interaction with Gaussian probability distribution and $k<=m$ in the limit $l$ to 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 $k<<m<<l$, 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 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 $k<<m<<l$.

Comments: 44 pages, 3 postscript figures, revised version including a new proof of one of our main claims