Quantized topological phases beyond square lattices in Floquet synthetic dimensions

Topological effects manifest in a variety of lattice geometries. While square lattices, due to their simplicity, have been used for models supporting nontrivial topology, several exotic topological phenomena such as Dirac points, Weyl points and Haldane phases are most commonly supported by non-square lattices. Examples of prototypical non-square lattices include the honeycomb lattice of graphene and the Kagome lattice, both of which break fundamental symmetries and can exhibit quantized transport, especially when long-range hoppings and gauge fields are incorporated. The challenge of controllably realizing long-range hoppings and gauge fields has motivated a large body of research focused on harnessing lattices encoded in "synthetic" dimensions. Photons in particular have many internal degrees of freedom and hence show promise for implementing these synthetic dimensions; however, most photonic synthetic dimensions has hitherto created 1D or 2D square lattices. Here we show that non-square lattice Hamiltonians can be implemented using Floquet synthetic dimensions. Our construction uses dynamically modulated ring resonators and provides the capacity for direct k-space engineering of lattice Hamiltonians. Such a construction lifts constraints on the orthogonality of lattice vectors that make square geometries simpler to implement, and instead transfers the complexity to the engineering of complex Floquet drive signals. We simulate topological signatures of the Haldane and the brick-wall Haldane model and observe them to be robust in the presence of external optical drive and photon loss, and discuss unique characteristics of their topological transport when implemented on these Floquet lattices. Our proposal demonstrates the potential of driven-dissipative Floquet synthetic dimensions as a new architecture for k-space Hamiltonian simulation of high-dimensional lattice geometries.