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Physics Letters B

Time translational symmetry

M. Lehto, HOLGER BECH NIELSEN, Masao Ninomiya C

Published Mar 1, 1989• 11 citations
Publisher
DOI
10.1016/0370-2693(89)90843-5

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Determinism and Time Symmetries

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Abstract We develop the effective field theoretical (EFT) approach to time-translational symmetry breaking of nonequilibrium open systems based on the Schwinger-Keldysh formalism. In the Schwinger-Keldysh formalism, all the symmetries of the microscopic Lagrangian are doubled essentially because the dynamical fields are doubled to describe the time-evolution along the closed-time-path. The effective Lagrangian for open systems are then obtained by coarse-graining the microscopic Schwinger-Keldysh Lagrangian. As a consequence of coarse-graining procedure, there appear the noise and dissipation effects, which explicitly break the doubled time-translational symmetries into a diagonal one. We therefore need to incorporate this symmetry structure to construct the EFT for Nambu-Goldstone bosons in symmetry broken phases of open systems. Based on this observation together with the consistency of the Schwinger-Keldysh action, we construct and study the general EFT for time-translational symmetry breaking in particular, having in mind applications to synchronization, time crystal, and cosmic inflation.
Localization transitions represent a fundamental class of continuous phase transitions, yet they occur without any accompanying symmetry breaking. We resolve this by introducing the concept of dynamical translational symmetry (DTS), which is defined not by the Hamiltonian but by the long-time dynamics of local observables. Its order parameter, the time-averaged local translational contrast (TLTC), quantitatively diagnoses whether evolution restores or breaks translational equivalence. We demonstrate that the TLTC universally captures the Anderson localization transition, the many-body localization transition, and topological phase transitions, revealing that these disparate phenomena are unified by the emergent breaking of DTS. This work establishes a unified dynamical-symmetry framework for phases transitions beyond the equilibrium paradigm.