High-frequency expansion for Floquet prethermal phases with emergent symmetries: Application to time crystals and Floquet engineering

Prethermalization, where quasisteady states are realized in the intermediate long-time regime (prethermal regime), in periodically driven (Floquet) systems is an important phenomenon since it provides a platform for nontrivial Floquet many-body physics. In this Rapid Communication, we consider Floquet systems with dual energy scales: The Hamiltonian consists of two different terms whose amplitude is either comparable to or much smaller than the frequency. As a result, when the larger-amplitude drive induces a ${\mathbb{Z}}_N$ symmetry operation, we obtain the effective static Hamiltonian respecting the emergent ${\mathbb{Z}}_N$ symmetry in high-frequency expansions, which describes the dynamics of such Floquet systems in the prethermal regime. As an application of our formulation, our formalism gives a general way to analyze time crystals in prethermal regimes in terms of the static effective Hamiltonian. We also provide an application to Floquet engineering, with which we can perform the simultaneous control of phases and symmetries of the systems. For example, this enables us to control symmetry protected topological phases even when the original system does not respect the symmetry.

Figure 1

(a) Deviation caused by the truncation of the high-frequency expansion for the model described by Eqs. (9) and (10). The number of the sites $L$ is four. (b) Intuitive picture of pDTCs. Until the prethermal regime ends, DTC orders are possibly realized.

Figure 2

Numerical results for the model described by Eqs. (9) and (10). Each of them is calculated by the exact diagonalization for the finite size $L=4$. (a), (b) Stroboscopic dynamics of the local magnetization $\langle \psi \left(t\right)|{\sigma }_i^z|\psi \left(t\right)\rangle$. The local magnetization oscillates with the period $2T$ in the long transient dynamics. (c) Lifetime of prethermal DTC orders. Here, the lifetime is defined by the time when the amplitude of the oscillation reaches 0.95 times as large as the first one.