Wigner--Dyson statistics for a class of integrable models

Abstract

We construct an ensemble of second--quantized Hamiltonians with two bosonic degrees of freedom, whose members display with probability one GOE or GUE statistics. Nevertheless, these Hamiltonians have a second integral of motion, namely the boson number, and thus are integrable. To construct this ensemble we use some ``reverse engineering'' starting from the fact that $n$--bosons in a two--level system with random interactions have an integrable classical limit by the old Heisenberg association of boson operators to actions and angles. By choosing an $n$--body random interaction and degenerate levels we end up with GOE or GUE Hamiltonians. Ergodicity of these ensembles completes the example.