  Critical points and phase transitions are characterized by diverging
susceptibilities, reflecting the tendency of the system toward spontaneous
symmetry breaking. Equilibrium statistical mechanics bounds these instabilities
to occur at zero frequency, giving rise to static order parameters. In this
work we introduce a new class of dynamical transitions in a quantum many body
system far from thermal equilibrium, characterized by a susceptibility
diverging at a finite non-zero frequency, an emerging scale set by interactions
and non-equilibrium effects. In the broken-symmetry phase the corresponding
macroscopic order parameter becomes non-stationary and oscillates in time
without damping, thus breaking continuous time-translational symmetry. Our
results, obtained for a paradigmatic model of bosons interacting on lattice in
prensence of drive and dissipation, are relevant for the upcoming generation of
circuit QED arrays experiments and outline a generic framework to study
time-domain instabilities in non-equilibrium quantum systems, including Floquet
time crystals and quantum synchronization.
