Condensed Matter Physics in Time Crystals

Abstract

Time crystals are physical systems whose time translation symmetry is spontaneously broken. Although the spontaneous breaking of continuous timetranslation 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.

Floquet theory for temporal correlations and spectra in time-periodic open quantum systems: Application to squeezed parametric oscillation beyond the rotating-wave approximation

“…In contrast, the study of quantum dynamics around limit cycles is a very active field of theoretical research in different contexts [45][46][47][48][49][50][51][52], including their connection to the emergence of time crystals [53][54][55][56] in driven-dissipative many-body systems [57][58][59][60][61][62][63][64]. Closely related to the latter is the field of Floquet or discrete time crystals in periodically-driven closed many-body systems [56,[65][66][67][68], which have only recently been identified as unconventional phases of matter far from equilibrium [69][70][71][72][73], but have already sparked interesting experiments [74][75][76][77][78][79][80][81] and applications [68,82]. Also in this context, periodic modulations allow for the so-called Floquet engineering of Hamiltonians [83], leading to some desired properties such as nontrivial topology or optimized transport [84].…”

Lack of a genuine time crystal in a chiral soliton model

“…On the other hand, another kind of time crystals has been demonstrated in the laboratory [12][13][14][15][16], i.e., the so-called discrete or Floquet time crystals where discrete time translation symmetry is spontaneously broken into another discrete time translation symmetry [17][18][19]. Discrete time crystals and condensed matter physics in the time domain are becoming intensively developing research area (for comprehensive reviews see [2,67,68]) but hunting for a genuine time crystal, which can be realized in the laboratory, is still continued.…”