Time Glasses: Symmetry Broken Chaotic Phase with a Finite Gap

TL;DR

This paper introduces the concept of a time glass, a phase in driven dissipative quantum systems characterized by long-range order and chaotic oscillations, with a finite spectral gap that persists in the thermodynamic limit.

Contribution

The paper defines the time glass phase, analyzes its spectral properties, and establishes a link between microscopic spectral gaps and macroscopic dynamics in quantum many-body systems.

Findings

Liouvillian gap remains finite in the thermodynamic limit in the time glass phase.

Spectral gap correlates with the decay rate of order-parameter autocorrelation.

Long-lived transients exist despite a finite spectral gap due to quantum Rényi divergence growth.

Abstract

We introduce the time glass, a non-periodic analogue of the discrete time crystal that arises in periodically driven dissipative quantum many-body systems. This phase is defined by two key features: (i) spatial long-range order arising from the spontaneous breaking of an internal symmetry, and (ii) temporally chaotic oscillations of the order parameter, whose lifetime diverges with system size. In other words, a time glass is a state of matter in which all components evolve in a synchronized yet chaotic manner. To characterize the time glass phase, we focus on the spectral gap of the one-cycle (Floquet) Liouvillian, which determines the decay rate of the slowest relaxation mode. Numerical studies of periodically driven dissipative Ising models show that, in the time glass phase, the Liouvillian gap remains finite in the thermodynamic limit, in contrast to time crystals where the gap closes exponentially with system size. We further demonstrate that the Liouvillian gap converges to the decay rate of the order-parameter autocorrelation derived from the classical (mean-field) dynamics in the thermodynamic limit. This result establishes a direct correspondence between microscopic spectral features and emergent macroscopic dynamics in driven dissipative quantum systems. At first glance, the existence of a nonzero Liouvillian gap appears incompatible with the presence of indefinitely persistent chaotic oscillations. We resolve this apparent paradox by showing that the quantum R\'enyi divergence between a localized coherent initial state and the highly delocalized steady state grows unboundedly with system size. This divergence allows long-lived transients to persist even in the presence of a finite Liouvillian gap.