Sufficient condition for gapless spin-boson Lindbladians, and its connection to dissipative time-crystals

TL;DR

This paper establishes a condition linking steady-state correlations to gapless modes in Lindbladian systems, revealing how such modes can induce persistent dynamics and dissipative time-crystals in collective spin-boson models.

Contribution

It introduces a sufficient condition for gapless excitations in Lindbladians and connects this to the emergence of dissipative time-crystals, supported by analytical proofs and model studies.

Findings

Nonzero steady-state cumulants imply gapless Lindbladian modes.

Gapless modes can cause persistent oscillations in spin observables.

Mean-field approximation is exact in these systems under certain conditions.

Abstract

We discuss a sufficient condition for gapless excitations in the Lindbladian master equation for collective spin-boson systems and permutationally invariant systems. The condition relates a nonzero macroscopic cumulant correlation in the steady state to the presence of gapless modes in the Lindbladian. In phases arising from competing coherent and dissipative Lindbladian terms, we argue that such gapless modes, concomitant with angular momentum conservation, can lead to persistent dynamics in the spin observables with the possible formation of dissipative time-crystals. We study different models within this perspective, from Lindbladians with Hermitian jump operators, to non-Hermitian ones composed by collective spins and Floquet spin-boson systems. We also provide a simple analytical proof for the exactness of mean-field semiclassical approach in such systems based on a cumulant expansion.