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

TL;DR

This paper explores how systems with quasi-ODLRO exhibit long-lived time-dependent order parameters, revealing spontaneous breaking of time reversal symmetry in macroscopic quantum states with nearly conserved charge Q.

Contribution

It clarifies the conditions under which time translation symmetry is broken in systems with quasi-ODLRO, connecting relaxation times to observable symmetry breaking.

Findings

Systems with LRO have long-lived time-dependent order parameters.

Time reversal symmetry breaking is observable when charge Q is nearly conserved.

Examples include superfluid 4He, magnon BEC, and precessing vortices.

Abstract

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.