Hilbert-space geometry of quantum chaos

The quantum geometric tensor (QGT) characterizes the Hilbert-space geometry of eigenstates of parameter-dependent Hamiltonian. The real symmetric part of QGT is proportional to the Riemann geometric tensor on the space of quantum states while its imaginary antisymmetric part gives rise to the geometric (Berry) curvature. We consider the symmetric part of the QGT for different multi-parametric random matrix models and discuss the nature of ergodic-nonergodic transition from the point of view of quantum geometry. We found that while the ergodic phase corresponds to the smooth quantum geometry, the integrable limit marks itself as a singular geometry.