  Ensembles of quantum chaotic systems are expected to exhibit energy
eigenvalues with random-matrix-like level repulsion between pairs of energies
separated by less than the inverse Thouless time. Recent research has shown
that exact and approximate global symmetries of a system have clear signatures
in these spectral statistics, enhancing the spectral form factor or
correspondingly weakening level repulsion. This paper extends those results to
the case of spontaneous symmetry breaking, and shows that, surprisingly,
spontaneously breaking a symmetry further enhances the spectral form factor.
For both RMT-inspired toy models and models where the symmetry breaking has a
description in terms of fluctuating hydrodynamics, we obtain formulas for this
enhancement for arbitrary symmetry breaking patterns, including $Z_n$, $U(1)$,
and partially or fully broken non-Abelian symmetries.
