Stochastic properties of many-body systems

The k-body embedded ensembles of random matrices originally defined by Mon and French are investigated as paradigmatic models of stochasticity in Fermionic many-body systems. In these ensembles, m Fermions in l degenerate single-particle states, interact via a random k-body interaction which obeys unitary or orthogonal symmetry. We focus attention on the spectral properties of these ensembles. We always take the limit l→∞. For 2k>m, the spectral properties of the k-body embedded unitary and orthogonal ensembles coincide with those of the canonical Gaussian unitary and orthogonal random-matrix ensemble, respectively. For k⪡m⪡l, the spectral fluctuations become Poissonian. The reason for this behavior is displayed by constructing limiting ensembles. The case of embedded Bosonic ensembles (m Bosons in l degenerate single-particle states interact via a random k-body interaction which obeys unitary or orthogonal symmetry) is also considered and compared with the case of Fermions.