# 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.