Abelian Floquet symmetry-protected topological phases in one dimension

Time-dependent systems have recently been shown to support novel types of topological order that cannot be realized in static systems. In this paper we consider a range of time-dependent, interacting systems in one dimension that are protected by an Abelian symmetry group. We classify the distinct topological phases that can exist in this setting and find that they may be described by a bulk invariant associated with the unitary evolution of the closed system. In the open system, nontrivial phases correspond to the appearance of edge modes, which have signatures in the many-body quasienergy spectrum and which relate to the bulk invariant through a form of bulk-edge correspondence. We introduce simple models which realize nontrivial dynamical phases in a number of cases, and outline a loop construction that can be used to generate such phases more generally.

Figure 1

Schematic splitting of a Majorana multiplet at a $(0,\pi )$-phase transition as the system is closed with (a) periodic and (b) antiperiodic boundary conditions. Blue lines correspond to states with even parity, while red lines correspond to states with odd parity. Spectrum (b) can be obtained from spectrum (a) through a shift by $\pi$. See main text for details.