Hyperbolic Topological Band Insulators

Recently, hyperbolic lattices that tile the negatively curved hyperbolic plane emerged as a new paradigm of synthetic matter, and their energy levels were characterized by a band structure in a four- (or higher-) dimensional momentum space. To explore the uncharted topological aspects arising in hyperbolic band theory, we here introduce elementary models of hyperbolic topological band insulators: the hyperbolic Haldane model and the hyperbolic Kane-Mele model; both obtained by replacing the hexagonal cells of their Euclidean counterparts by octagons. Their nontrivial topology is revealed by computing topological invariants in both position and momentum space. The bulk-boundary correspondence is evidenced by comparing bulk and boundary density of states, by modeling propagation of edge excitations, and by their robustness against disorder.

Figure 1

Nearest-neighbor model on the $\{8,3\}$ lattice. (a) The lattice consists of octagons with coordination number three. The Bolza cell (multicolored octagon), the fundamental tile of the hyperbolic Bravais lattice, contains six elementary octagons and 16 sites (black and white dots). The colored arrows labeled ${\gamma }_1,\dots ,{\gamma }_4$ are the generators of the hyperbolic Bravais lattice. Rotation $R$ (dark red) and reflection $S$ (dark blue) are symmetries of the model. (b),(c) Bulk density of states $\rho (E)$ for the nearest-neighbor model on the $\{8,3\}$-lattice, extracted from hyperbolic band theory (HBT, red) vs. exact diagonalization (ED, blue) in the absence (b) vs presence (c) of a sublattice potential $M$.

Figure 2

Hyperbolic Haldane model. (a) Schematic depiction of the model. Red dashed lines indicate next-to-nearest neighbor hopping with amplitude $t_2{e}^{\pm \mathrm{i}\mathrm{\Phi }}$. The phases $\pm \mathrm{\Phi }$ arise due to alternating magnetic fluxes [symbols $\odot$ ($\otimes$) in the innermost octagon] through the system. (b) Density of states for $t_2=M/2=\frac{1}{6}$ computed from HBT, revealing three energy gaps. (c) Bulk-DOS functions ${\rho }^{\mathrm{HBT}}(E)$ and ${\rho }_{\mathrm{bulk}}^{\mathrm{ED}}(E)$ for $\mathrm{\Phi }=\pi /2$ (red dashed line in b) computed using HBT (red) and ED (blue), and the boundary-DOS ${\rho }_{\text{boundary}}^{\mathrm{ED}}$ (green).

Figure 3

Hyperbolic Kane-Mele model. (a) Density of states for $t_2=M/2=\frac{1}{6}$ computed from HBT, which reveals three energy gaps at small Rashba coupling ${\lambda }_R$. (b) Bulk-DOS functions for ${\lambda }_R=-\frac{1}{6}$ (red dashed line in (a) computed using HBT (red) and ED (blue), and the boundary-DOS function (green).

Figure 4

Topological edge states. (a) Edge-dispersion $E$ as a function of angular momentum $\ell$ for chiral edge states in the upper energy gap of the hyperbolic Haldane model. (b) Propagation of a Gaussian wave packet along the flake boundary; $⟨\alpha ⟩$ is the angular displacement in time $\tau$, and error bars indicate the width of the wave packet. (c) Snapshots of the wave packet at $\tau =0$, 240, 480, 720 (colored, respectively, red, yellow, green, and blue). The area of a disk centered at a given site encodes the probability to find the particle at that site.

Figure 5

Robustness against Anderson localization. For the reduced hyperbolic KM model, we consider inclusion of random spin-mixing terms that preserve (TRS, blue) or break (TRB, red) time-reversal symmetry. A dot with coordinates $(E_j,{\mathrm{IPR}}_j)$ represents the energy and the inverse participation ratio of an eigenstate $|{\varphi }_j⟩$ in the flake geometry. The random terms are drawn from a uniform distribution with $W_{\max }=0.02$ (a) and $W_{\max }=0.2$ (b). The blue (yellow) backgrounds indicate energy ranges corresponding to bulk gap (bulk band) in the absence of disorder.