# Floquet engineering of topological phases protected by emergent symmetries under resonant drives

## TL;DR

This paper proposes a method to realize and control nontrivial topological phases protected by emergent symmetries in Floquet systems using resonant drives, enabling switchable SPT phases and real-time topological diagnostics.

## Contribution

It introduces a novel Floquet engineering approach combining high-frequency and resonant drives to induce emergent symmetries and topological phases in nonequilibrium systems.

## Key findings

- Emergent $ extbf{Z}_2$ symmetry enables nontrivial SPT phases.
- Topological phases are switchable by static transverse field direction.
- Period doubling of nonlocal order parameter indicates topological phase.

## Abstract

Floquet engineering is one of the most vigorous fields in periodically driven (Floquet) systems, with which we can control phases of matter usually by high-frequency drives. In this paper, with Floquet engineering by a combination of high-frequency drives and resonant drives, we propose a way to realize nontrivial topological phases protected by a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry only in the presence of a $\mathbb{Z}_2$ symmetry, using a robust emergent $\mathbb{Z}_2$ symmetry induced by the resonant drives. Moreover, the symmetry protected topological (SPT) phases are switchable between nontrivial and trivial phases only by the direction of a static transverse field, and even perturbations on the resonant drive can be utilized to realize richer SPT phases. We also discuss the real-time dynamics of the model, and find that which topological phases the system lies in can be distinguished by a period doubling of a nonlocal order parameter, as with discrete time crystals. A realization or a control of nontrivial SPT phases without the required symmetries by resonant drives, proposed in this paper, would shed a new light on the observation of topological phenomena in nonequilibrium setups.