# Floquet time crystal in the Lipkin-Meshkov-Glick model

## TL;DR

This paper demonstrates the existence of a Floquet time crystal in the Lipkin-Meshkov-Glick model, showing its robustness, spectral properties, and potential for experimental realization with trapped ions.

## Contribution

It provides the first detailed analysis of Floquet time crystals in an infinite-range interacting spin system, linking symmetry breaking to time-translation symmetry breaking.

## Key findings

- Existence of Floquet time crystal linked to $ ext{Z}_2$ symmetry breaking.
- Identification of Floquet doublets with quasi-energies differing by half the driving frequency.
- Robustness of the time crystal under perturbations and potential experimental implementation.

## Abstract

In this work we discuss the existence of time-translation symmetry breaking in a kicked infinite-range-interacting clean spin system described by the Lipkin-Meshkov-Glick model. This Floquet time crystal is robust under perturbations of the kicking protocol, its existence being intimately linked to the underlying $\mathbb{Z}_2$ symmetry breaking of the time-independent model. We show that the model being infinite-range and having an extensive amount of symmetry breaking eigenstates is essential for having the time-crystal behaviour. In particular we discuss the properties of the Floquet spectrum, and show the existence of doublets of Floquet states which are respectively even and odd superposition of symmetry broken states and have quasi-energies differing of half the driving frequencies, a key essence of Floquet time crystals. Remarkably, the stability of the time-crystal phase can be directly analysed in the limit of infinite size, discussing the properties of the corresponding classical phase space. Through a detailed analysis of the robustness of the time crystal to various perturbations we are able to map the corresponding phase diagram. We finally discuss the possibility of an experimental implementation by means of trapped ions.