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 reversal 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 >> \tau_E, the perturbed system relaxes first to the state with the ODLRO, which persists for a long time \tau_Q and finally relaxes to the full equilibrium static state. In the limit \tau_Q -> \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 reversal symmetry. Examples of systems with quasi-ODLRO are provided by superfluid phase of liquid 4He, Bose-Einstein condensation of magnons (phase coherent spin precession) and precessing vortices.