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 the bulk, but quantized energy currents at edge. These were discovered through an abstract classification, and few microscopic models exist. We develop a coupled layer construction for tight-binding of these $d\ensuremath{\in}{1,2}$ spatial dimensions, with any number incommensurate drive frequencies $D$. exhibit responses associated synthetic two- four-dimensional quantum Hall effects steady state. A numerical study phase diagram $(d+D)=(1+2)$ shows (i) robust trivial separated by sharp transition, (ii) charge diffusion half-integer current between drives (iii) long-lived which remains present when weak interactions introduced.