Abstract
We establish the existence of ‘time quasilattices’ as stable trajectories in dissipative dynamical systems. These tilings of the time axis, with two unit cells of different durations, can be generated as cuts through a periodic lattice spanned by two orthogonal directions of time. We show that there are precisely two admissible time quasilattices, which we term the infinite Pell and Clapeyron words, reached by a generalization of the period-doubling cascade. Finite Pell and Clapeyron words of increasing length provide systematic periodic approximations to time quasilattices which can be verified experimentally. The results apply to all systems featuring the universal sequence of periodic windows. We provide examples of discrete-time maps, and periodically-driven continuous-time dynamical systems. We identify quantum many-body systems in which time quasilattices develop rigidity via the interaction of many degrees of freedom, thus constituting dissipative discrete ‘time quasicrystals’.
Highlights
The only break which we make with this convention is in Section 6 in which we identify time quasilattices stabilized by many-body interactions: in order to emphasize that these states constitute an extension of the concept of time crystals to include quasilattice symmetry, we term them time quasicrystals, despite the fact that they exist in one dimension of time
In this paper we have demonstrated the existence of time quasilattices in dissipative dynamical systems
We demonstrated that time quasilattices can appear as stable, attracting orbits in any dissipative nonlinear dynamical system which features the universal sequence, or any re-ordering thereof [34, 39]
Summary
The spontaneous breaking of translation symmetry occurs whenever a crystal grows from a liquid. It later transpired that such a process is impossible [4,5,6], a loophole left open the possibility of breaking the discrete time translation symmetry of periodically-driven systems down to a multiple of the period [7,8,9,10] This led to physical implementations in both cold atoms and nitrogen vacancy defects in diamond [11, 12].
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