# Quantum chaos for non-standard symmetry classes in the Feingold-Peres model of coupled tops. (arXiv:1709.03584v1 [quant-ph])

We consider two coupled quantum tops with angular momentum vectors
$\mathbf{L}$ and $\mathbf{M}$. The coupling Hamiltonian defines the
Feinberg-Peres model which is a known paradigm of quantum chaos. We show that
this model has a non-standard symmetry with respect to the Altland-Zirnbauer
ten-fold symmetry classification of quantum systems which extends the
well-known three-fold way of Wigner and Dyson (referred to as `standard'
symmetry classes here). We identify that the non-standard symmetry classes
BD$I_0$ (chiral orthogonal class with no zero modes), BD$I_1$ (chiral
orthogonal class with one zero mode) and C$I$ (anti-chiral orthogonal class) as
well as the standard symmetry class A$I$ (orthogonal class). We numerically
analyze the specific spectral quantum signatures of chaos related to the
non-standard symmetries. In the microscopic density of states and in the
distribution of the lowest positive energy eigenvalue we show that the
Feinberg-Peres model follows the predictions of the Gaussian ensembles of
random-matrix theory in the appropriate symmetry class if the corresponding
classical dynamics is chaotic. In a crossover to mixed and near-integrable
classical dynamics we show that these signatures disappear or strongly change.