Chern insulator in a hyperbolic lattice

TL;DR

This paper demonstrates the existence of Chern insulator phases with quantized conductance and chiral edge states in a hyperbolic 8,3 lattice, revealing robustness against disorder and discovering a non-Euclidean topological Anderson insulator.

Contribution

The study introduces the first demonstration of Chern insulators in hyperbolic lattices, including the effects of disorder and the emergence of a hyperbolic topological Anderson insulator.

Findings

Identification of two Chern insulator phases with opposite chirality.

Quantized conductance plateaus confirmed by local current distribution.

Disorder induces topological phases, including a hyperbolic topological Anderson insulator.

Abstract

Motivated by the recent experimental realizations of hyperbolic lattices in circuit quantum electrodynamics and the research interest in the non-Euclidean generalization of topological phenomena, we investigate the Chern insulator phases in a hyperbolic $\{8,3\}$ lattice, which is made from regular octagons ($8$-gons) such that the coordination number of each lattice site is $3$. Based on the conformal projection of the hyperbolic lattice into the Euclidean plane, i.e., the Poincar\'{e} disk model, by calculating the Bott index ($B$) and the two-terminal conductance, we reveal two Chern insulator phases (with $B=1$ and $B=-1$, respectively) accompanied with quantized conductance plateaus in the hyperbolic $\{8,3\}$ lattice. The numerical calculation results of the nonequilibrium local current distribution further confirm that the quantized conductance plateau originates from the chiral edge states and the two Chern insulator phases exhibit opposite chirality. Moreover, we explore the effect of disorder on topological phases in the hyperbolic lattice. It is demonstrated that the chiral edge states of Chern insulators are robust against weak disorder in the hyperbolic lattice. More fascinating is the discovery of disorder-induced topological non-trivial phases exhibiting chiral edge states in the hyperbolic lattice, realizing a non-Euclidean analog of topological Anderson insulator. Our work provides a route for the exploration of topological non-trivial states in hyperbolic geometric systems.