# High-frequency Expansion for Floquet Prethermal Phases with Emergent Symmetries: Application to Time Crystals and Floquet Engineering

## TL;DR

This paper develops a high-frequency expansion method to analyze Floquet prethermal phases with emergent symmetries, enabling the study of time crystals and Floquet engineering with controlled phases and symmetries.

## Contribution

It introduces a novel high-frequency expansion framework for Floquet systems with dual energy scales, revealing emergent symmetries and enabling new control over phases and topological properties.

## Key findings

- Effective static Hamiltonian with emergent _N_N symmetry derived
- Framework applied to analyze prethermal discrete time crystals
- Demonstrated control of symmetry-protected topological phases

## Abstract

Prethermalization, where quasi-steady 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 of nontrivial Floquet many-body physics. In this Letter, we consider Floquet systems with dual energy scales: the Hamiltonian consists of two different terms whose amplitude is either comparable 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 a new 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, we consider prethermal discrete time crystals, in which our formalism gives a general way to analyze them in the prethermal regime in terms of the static effective Hamiltonian. We also provide an application to Floquet engineering, with which we can perform simultaneous control of phases and symmetries of the systems. This enables us to control symmetry protected topological phases even when the original system does not respect the symmetry.