# Time-crystalline behavior in an engineered spin chain

## TL;DR

This paper demonstrates that a finite polarized XX spin chain can exhibit time-crystalline behavior with equidistant quasienergy spectra under periodic driving, without requiring disorder or interactions, and for all initial states.

## Contribution

It introduces a clean, interaction-free system that displays time crystallization, expanding understanding of conditions necessary for time crystal formation.

## Key findings

- The spin chain maintains equidistant quasienergies under various driving schemes.
- Time crystals of different periodicities are realized for all initial states.
- The system's robustness against parameter variations is confirmed.

## Abstract

Time crystals appear when systems display a commensurate spontaneous breaking of the discrete time translational invariance imposed by an external periodic drive. No consensus on the definition has been reached as yet, but important aspects comprise robustness against small variations of the parameters and the initial quantum state. Often, disorder and interaction are thought to be essential ingredients for the occurrence of time crystals. We study a finite-length polarized XX spin chain engineered to display a spectrum of equidistant energy levels without drive and show that it keeps a spectrum of equidistant quasienergies in Floquet theory for a large variety of periodic driving schemes. This interesting behavior is explained by mapping the XX spin chain with $N+1$ sites to a single large spin with $S=N/2$ invoking the closure of the group SU(2). For suitably tuned parameters this system realizes time crystals of various periodicities for \emph{all} initial states. The robustness against variations of the parameters is also discussed. Thereby, we establish a clean system without interaction which can display the phenomenon of time crystallization.