Topological Hyperbolic Lattices

Non-Euclidean geometry, discovered by negating Euclid’s parallel postulate, has been of considerable interest in mathematics and related fields for the description of geographical coordinates, Internet infrastructures, and the general theory of relativity. Notably, an infinite number of regular tessellations in hyperbolic geometry—hyperbolic lattices—are expected to extend Euclidean Bravais lattices and the consequent wave phenomena to non-Euclidean geometry. However, topological states of matter in hyperbolic lattices have yet to be reported. Here we investigate topological phenomena in hyperbolic geometry, exploring how the quantized curvature and edge dominance of the geometry affect topological phases. We report a recipe for the construction of a Euclidean photonic platform that inherits the topological band properties of a hyperbolic lattice under a uniform, pseudospin-dependent magnetic field, realizing a non-Euclidean analog of the quantum spin Hall effect. For hyperbolic lattices with different quantized curvatures, we examine the topological protection of helical edge states and generalize Hofstadter’s butterfly, by employing two empirical parameters that measure the edge confinement and defect immunity. We demonstrate that the proposed platforms exhibit the unique spectral-magnetic sensitivity of topological immunity in highly curved hyperbolic planes. Our approach is applicable to general non-Euclidean geometry and enables the exploitation of infinite lattice degrees of freedom for band theory.

Figure 1

Hyperbolic lattices. (a)–(c) (left) The saddle-shaped surface of the hyperbolic plane and (right) Poincaré disks for different $K$. (d) Recursive generation of the hyperbolic lattice $\{4,6\}$. (e) A graph view of the hyperbolic lattice. For the original vertex positions in the polar coordinate ($r$, $\xi$), each deformed graph is obtained with $f(r)=r^2$ and $g(\xi )={\xi }^2/(2\pi )$.

Figure 2

Photonic hyperbolic lattices. (a)–(c) Coupling schematics for the identical connection between lattice sites. (d) An example of photonic hyperbolic lattices: $\{4,6\}$ lattice at the epoch 2.

Figure 3

Hyperbolic QSHE. (a)–(c) Hyperbolic gauge for a uniform magnetic field: (a) seed polygon, (b) polygons in the next epoch, and (c) the final result. The blue and green arrows in (a) and (b) denote the gauge field assigned in the first and second epoch, respectively. ${\phi }_{q,p}$ in (c) represents the gauge field from the $p$th to $q$th element. (d)–(f) $C_{\text{edge}}^{(n)}$ as a function of $\omega$ and $\theta$: (d) Euclidean lattice with $K=0$, and (e),(f) hyperbolic lattices of (e) $K=-1.26$ and (f) $K=-2.09$. ${\omega }_0=1$. All other parameters are the same as those in Fig. 2.

Figure 4

Topological protection and Hofstadter’s “hyperbolic” butterflies. (a),(b) Transmission through topologically protected edge states in the lattice $\{4,5\}$: (a) backward ($t=99.50\%$) and (b) forward ($t=98.51\%$) transmission [black arrows in (c)]. Different radii of the elements represent diagonal disorder. (c) Transmission spectrum at $B=\theta =0.9\pi$ (red line, average; gray regions, standard deviation). (d)–(f) $C_{\text{immune}}$ for (d) Euclidean and (e),(f) hyperbolic lattices. For the cases in (c)–(f), we examine an ensemble of 50 realizations of disorder. The case in (c) denotes the white dashed line in (e). $\mathrm{\Delta }={\omega }_0/200$ for all cases. The coupling between the boundary elements and the input or output waveguide is $0.08{\omega }_0$. All other parameters are the same as those in Fig. 3.