Abstract
Time crystals are physical systems whose time translation symmetry is spontaneously broken. Although the spontaneous breaking of continuous time-translation symmetry in static systems is proved impossible for the equilibrium state, the discrete time-translation symmetry in periodically driven (Floquet) systems is allowed to be spontaneously broken, resulting in the so-called Floquet or discrete time crystals. While most works so far searching for time crystals focus on the symmetry breaking process and the possible stabilising mechanisms, the many-body physics from the interplay of symmetry-broken states, which we call the condensed matter physics in time crystals, is not fully explored yet. This review aims to summarise the very preliminary results in this new research field with an analogous structure of condensed matter theory in solids. The whole theory is built on a hidden symmetry in time crystals, i.e., the phase space lattice symmetry, which allows us to develop the band theory, topology and strongly correlated models in phase space lattice. In the end, we outline the possible topics and directions for the future research.
Highlights
These two inequalities consist of the ‘no-go theorem’, which strictly prohibits the existence of spontaneously rotating time crystals for the ground state and thermal equilibrium state
We started from Wilczek’s time crystals breaking the continuous time-translation symmetry (CTTS) of a static Hamiltonian [1,2], which has been disproved to exist in equilibrium systems [5,8,9,16,18]
In periodically driven (Floquet) systems, the time crystal behaviour does exist by breaking the discrete time-translation symmetry (DTTS) [20,21,22,23,24], namely, the system responds at a fraction ωd/n of the original driving frequency
Summary
Keywords: Floquet time crystals, discrete time crystals, phase space crystals, Floquet engineering, symmetry breaking, condensed matter, noncommutative geometry Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
