Simultaneous symmetry breaking in spontaneous Floquet states: temporal Floquet-Nambu-Goldstone modes, Floquet thermodynamics, and the time operator

TL;DR

This paper explores symmetry breaking in Floquet states, introducing Floquet-Nambu-Goldstone modes, a time operator concept, and thermodynamic descriptions, with applications to a time supersolid and experimental proposals.

Contribution

It develops a formalism for symmetry breaking in Floquet states, including the quantization of Floquet-Nambu-Goldstone modes and the introduction of a time operator in quantum mechanics.

Findings

Identification of a temporal Floquet-Nambu-Goldstone mode in spontaneous Floquet states.

Numerical confirmation of the dominance of the temporal FNG mode in density correlations.

Proposal of an experimental scheme to observe the temporal FNG mode.

Abstract

We study simultaneous symmetry breaking in spontaneous Floquet states, focusing on the specific case of an atomic condensate. We first describe the quantization of the Nambu-Goldstone (NG) modes for a stationary state simultaneously breaking several symmetries of the Hamiltonian by invoking the generalized Gibbs ensemble, which enables a thermodynamical description of the problem. The quantization procedure involves a Berry-Gibbs connection, which depends on the macroscopic conserved charges associated to each broken symmetry and whose curvature is not invariant under generalized gauge transformations. We extend the formalism to Floquet states, where Goldstone theorem translates into the emergence of Floquet-Nambu-Goldstone (FNG) modes with zero quasi-energy. In the case of a spontaneous Floquet state, there is a genuine temporal FNG mode arising from the continuous time-translation symmetry breaking, whose quantum amplitude provides a rare realization of a time operator in Quantum Mechanics. Furthermore, since they conserve energy, spontaneous Floquet states can be shown to possess a conserved Floquet charge. Conventional Floquet systems also admit a thermodynamic description in terms of the Floquet enthalpy, the Legendre transform of the energy with respect to the Floquet charge, as these operate at fixed frequency. We apply our formalism to a particular realization of spontaneous Floquet state, the CES state, which breaks $U(1)$ and time-translation symmetries, representing a time supersolid. We numerically compute its density-density correlations, predicted to be dominated by the temporal FNG mode at long times, observing a remarkable agreement between simulation and theory. Based on these results, we propose a feasible experimental scheme to observe the temporal FNG mode of the CES state.