|VOLUME 98 | ISSUE 8 |
On the broken time translation symmetry in macroscopic systems: precessing states and off-diagonal long-range order
G. E. Volovik
Low Temperature Laboratory, Aalto University, School of Science and Technology, P.O. Box 15100, FI-00076 AALTO, Finland
Landau Institute for Theoretical Physics of the RAS, 119334 Moscow, Russia
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
These systems have two well separated relaxation times: the relaxation time τQ of
the charge Q and the energy relaxation time τE. If , 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 ,
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
4He, Bose-Einstein condensation of magnons (phase coherent spin precession) and
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