Coupled Layer Construction for Synthetic Hall Effects in Driven Systems

Quasiperiodically driven fermionic systems can support topological phases not realized in equilibrium. The fermions are localized in the bulk, but support quantized energy currents at the edge. These phases were discovered through an abstract classification, and few microscopic models exist. We develop a coupled layer construction for tight-binding models of these phases in $d\in\{1,2\}$ spatial dimensions, with any number of incommensurate drive frequencies $D$. The models exhibit quantized responses associated with synthetic two- and four-dimensional quantum Hall effects in the steady state. A numerical study of the phase diagram for $(d+D) = (1+2)$ shows: (i) robust topological and trivial phases separated by a sharp phase transition; (ii) charge diffusion and a half-integer energy current between the drives at the transition; and (iii) a long-lived topological energy current which remains present when weak interactions are introduced.