Ergodic and Discrete Time Crystal Phases in Periodically Kicked Many-Body Quantum Systems: An Analytical Study

TL;DR

This paper analytically investigates the emergence of discrete time crystal phases in periodically kicked many-body quantum systems, revealing conditions for oscillatory behavior and long-time thermalization.

Contribution

It provides an analytical framework explaining how discrete time crystals arise in nonintegrable spin chains under periodic kicking.

Findings

Observables thermalize to infinite-temperature averages at long times.

Subharmonic oscillations indicate discrete time crystal phases.

The same system can be ergodic or exhibit time crystal behavior depending on initial states.

Abstract

We analytically study the time evolution of the expectation values of observables in periodically kicked many-body quantum systems. Starting from an initial state, we compute both the transient and the long-time properties of the observables. Our derivation explains the criteria and the mechanism that lead to the infinite-temperature statistical average of observables at long times, irrespective of the initial state. When the criteria are violated, the observables oscillate with time. These oscillations are subharmonic and robust to small perturbations, suggesting the emergence of a discrete time crystal phase. We demonstrate these features explicitly in periodically kicked nonintegrable spin chains. For a spin chain with two kicks per cycle, we show that the kicked chain can exhibit an ergodic or a discrete-time crystal phase for the same kicking strengths, depending on the initial state preparation. We complement our time-evolution study of observables with the spectral form factor of these kicked models.