Dynamically enriched topological orders in driven two-dimensional systems

Time-periodic driving of a quantum system can enable new dynamical topological phases of matter that could not exist in thermal equilibrium. We investigate two related classes of dynamical topological phenomena in 2D systems: Floquet symmetry-protected topological phases (FSPTs) and Floquet enriched topological orders (FETs). By constructing solvable lattice models for a complete set of 2D bosonic FSPT phases, we show that bosonic FSPTs can be understood as topological pumps which deposit loops of 1D SPT chains onto the boundary during each driving cycle, which protects a nontrivial edge state by dynamically tuning the edge to a self-dual point poised between the 1D SPT and trivial phases of the edge. By coupling these FSPT models to dynamical gauge fields, we construct solvable models of FET orders in which anyon excitations are dynamically transmuted into topologically distinct anyon types during each driving period. These bosonic FSPT and gauged FSPT models are classified by group cohomology methods. In addition, we also construct examples of “beyond cohomology” FET orders, which can be viewed as topological pumps of 1D topological chains formed of emergent anyonic quasiparticles.

Figure 1

Model: The spin model consists of Ising spins on a square lattice, with a two-site ($A$ and $B$) unit cell (dashed shaded oval in left panel). The sequence schematically depicts how locally pumping a short loop of 1D SPTs onto each plaquette results in toggling the edge between the trivial unentangled state and the 1D SPT phase. This dynamically tunes the edge to a self-dual point as described in the text.

Figure 2

Gauging the FSPT model: The global ${\mathbb{Z}}_2^A\times {\mathbb{Z}}_2^B$ symmetry can be promoted to a local gauge symmetry, by adding ${\mathbb{Z}}_2^A$ gauge link variables ${\alpha }_{ij}$ (blue dots) to the bonds connecting $A$ sites (large-dashed blue lines), and ${\mathbb{Z}}_2^B$ gauge link variables ${\beta }_{kl}$ (red dots) to the bonds connecting $B$ sites (small-dashed red lines). ${\mathbb{Z}}_2^{A/B}$ gauge fluxes reside on the tilted square plaquettes (shaded blue and red, respectively).

Figure 3

Dynamical anyon permutation: Schematic depiction of dynamical anyon permutation in a Floquet-enriched ${\mathbb{Z}}_2$ gauge theory, in which a topological superconducting chain (rounded red rectangle with arrows) of the $\psi =e\times m$ particles is pumped from the bulk onto the boundary during each period. Anyonic $e$ and $m$ excitations are permuted as they pass through the topological chain.