Hyperbolic Chern Insulators

The Haldane model transforms the semi-metallic graphene into a Chern insulator by introducing complex-valued hopping terms t₂ e^±iφ between the second-nearest neighbors. It is an open question whether the Chern phases are generally present in hyperbolic lattices, recently realized experimentally in circuit QED and topolectrical circuitry. In this work, we theoretically generalize the Haldane model to a large number of regular {p, q} hyperbolic lattices, tessellated by p-sided polygons with q polygons meeting at each site, satisfying (p − 2)(q − 2) > 4. We numerically generate the finite-sized lattices on the Poincaré disk, with p ranging from 6 to 10 and q from 3 to 5, on which we construct the tight-binding hyperbolic Haldane models. Induced gaps are observed in all {p,3} hyperbolic Haldane models, and the computation of real-space Chern numbers indicate nontrivial Chern phases at these gaps. As q increases from 3 to 5, eigenstates appear in the induced gaps, trivializing the real-space Chern numbers. We attribute the absence of nontrivial Chern phases in models with larger q to the stronger negative curvature, which hinders the formation of cyclotron orbits. Furthermore, we discover a universality in the density-of-states as a function of hopping phase φ, such that their qualitative features depend only on p.