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

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

2000-10-26

arxiv.org/abs/cond-mat/0010426v2

Annals Phys. 292 (2001) 67-94

Physics

Condensed Matter

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

Scientific paper

10.1006/aphy.2001.6156

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 $k << m << l$.

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