We present a new mechanism for realizing of time crystal phase with arbitrary period, termed as commensurate time crystal (CTC), in a quantum seesaw constructed by an interacting Bose-Hubbard model. By periodically modulating the Hamiltonian, we find that the wave function will recover to its initial condition to when the modulating frequency is commensurate with the initial energy level spacing between the ground and first excited levels. The period is determined by the driving frequency and commensurate ratio. In this case, the wave function will almost be restricted to the lowest two instantaneous energy levels. By projecting the wave function to these two relevant states, the dynamics is exactly the same as that for a spin precessing dynamics and nutating dynamics about an rotating axis. The commensurate motion between these two dynamics give rise to CTC. We map out the corresponding phase diagram and show that in the low frequency regime the state is thermalized and in the strong modulating limit, the dynamics is determined by the effective Floquet Hamiltonian. This CTC is realizable between these two limits. The measurement of this dynamics from mean position and mean momentum in phase space is also discussed.
In a recent publication [Phys. Rev. Lett. 124, 178902 (2020)], Öhberg and Wright claim that in a chiral soliton model it is possible to realize a genuine time crystal which corresponds to a periodic evolution of an inhomogeneous probability density in the lowest-energy state.We show that this result is incorrect and present a solution which possesses lower energy with the corresponding probability density that does not reveal any motion.It implies that the authors' conclusion that a genuine time crystal can exist in the system they consider is not true.
Abstract By analogy with the formation of space crystals, crystalline structures can also appear in the time domain. While in the case of space crystals we often ask about periodic arrangements of atoms in space at a moment of a detection, in time crystals the role of space and time is exchanged. That is, we fix a space point and ask if the probability density for detection of a system at this point behaves periodically in time. Here, we show that in periodically driven systems it is possible to realize topological insulators, which can be observed in time. The bulk-edge correspondence is related to the edge in time, where edge states localize. We focus on two examples: Su-Schrieffer-Heeger model in time and Bose Haldane insulator which emerges in the dynamics of a periodically driven many-body system.