Topological Edge Modes in a Floquet Hyperbolic System

Periodically driven (Floquet) systems can exhibit new topological properties absent in equilibrium systems. One example [1] is the existence of anomalous topological edge modes while the bulk bands have zero Chern numbers. In this work, we study a Floquet insulating system consisting of sites on a finite two-dimensional hyperbolic lattice that tessellates a finite portion of a negatively curved space. We show that in our Floquet system, while the bulk states have zero Chern numbers, there exists a chiral topological edge band with no analog in equilibrium topological insulators. Our study extends previous investigations [2, 3, 4] of topology in hyperbolic lattices. Due to the unique property of negatively curved spaces, a finite ratio of a hyperbolic lattice belongs to its boundary, even in the thermodynamic limit. As a result, the edge modes of a topological system on a hyperbolic lattice comprise a finite ratio of the total density of states. Unlike previous works, the topological edge modes in our system are achieved by a multi-step driving protocol without any need for complex hoppings, i.e., magnetic fluxes threaded through the lattice. Finally, we study the protection of the edge modes against disorder.

[1] Rudner, M. S., et al., Physical Review X 3.3 (2013): 031005

[2] Yu, S., et al., Physical Review Letters 125.5 (2020): 053901

[3] Liu, Z. R., et al., Physical Review B 105.24 (2022): 245301

[4] Urwyler, D. M., et al., arXiv preprint arXiv:2203.07292 (2022)