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.
Highlights
Among various applications of the effective field theoretical (EFT) approach, one interesting direction recently explored intensively is the application to real-time nonequilibrium dynamics of open systems, where the dynamics in interests is affected by the noise and dissipation originated from environments
We develop the effective field theoretical (EFT) approach to time-translational symmetry breaking of nonequilibrium open systems based on the Schwinger-Keldysh formalism
Regardless of our weak definition of spontaneous symmetry breaking (SSB), eq (2.39) plays an essential role since it provides a starting point to construct the EFT for timetranslational symmetry breaking in open systems; eq (2.39) together with e.g. a slow-roll approximation says that low-energy dynamics of non-stationary systems can be described by effective field theory of the Nambu-Goldstone field,6 which represents a perturbation around the time-dependent background
Summary
We elaborate on the symmetry structure attached to open systems by analyzing a simple example; that is, the Brownian particle system. We explain how the fluctuation dissipation relation can be incorporated as a consequence of a discrete symmetry. Readers familiar with the MSR formalism and the Schwinger-Keldysh formalism can skip most of this section after checking our weak definition of SSB provided in the end of section 2.2.1
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