On the broken time translation symmetry in macroscopic systems: precessing states and off-diagonal long-range order

The broken symmetry state with off-diagonal long-range order (ODLRO), which is characterized by the vacuum expectation value of the operator of creation of the conserved quantum number $Q$, has the time-dependent order parameter. However, the breaking of the time translation symmetry is observable only if the charge $Q$ is not strictly conserved and may decay. This dihotomy is resolved in systems with quasi-ODLRO. These systems have two well separated relaxation times: the relaxation time $\tau_Q$ of the charge $Q$ and the energy relaxation time $\tau_E$. If $\tau_Q \gg \tau_E$, the perturbed system relaxes first to the state with the ODLRO, which persists for a long time and finally relaxes to the full equilibrium static state. In the limit $\tau_Q \rightarrow \infty$, but not in the strict limit case when the charge $Q$ is conserved, the intermediate ODLRO state can be considered as the ground state of the system at fixed $Q$ with the observable spontaneously broken time translation symmetry. Examples of systems with quasi-ODLRO are provided by superfluid phase of liquid $^4$He, Bose–Einstein condensation of magnons (phase coherent spin precession) and precessing vortices.