Hyperbolic Topological Band Insulators

Abstract

The Bloch band theory describes energy levels of crystalline media by a collection of bands in momentum space. These bands can be characterized by non-trivial topological invariants, which via bulk-boundary correspondence imply protected boundary states inside the bulk energy gap. 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 non-trivial 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 modelling propagation of edge excitations, and by their robustness against disorder.

Flat bands and band touching from real-space topology in hyperbolic lattices

“…The key differences between Euclidean and non-Euclidean flat bands, as well as the precise fraction of states that lie in the flat band and the number of band-touching points (Table I), follow ultimately from the unusual higher-genus topology imposed by PBC in hyperbolic space [32,49], which is captured by our realspace and HBT arguments. The unique combination of real-space and (hyperbolic-)momentum-space characterization allows us to also capture certain concrete aspects of bulk eigenstates transforming according to non-Abelian irreps of the Fuchsian translation group [32]-a feat not achieved by prior works utilizing HBT [38,41]. Namely, we find that the fraction of the spectral weight lying in the flat band is the same for both Abelian and non-Abelian irreps of the Fuchsian translation group.…”

“…With recent experimental realizations in circuit quantum electrodynamics (cQED) [27] and electrical circuits [28], hyperbolic lattices are periodic in the non-Euclidean sense and correspond to regular tessellations of the hyperbolic plane [29], i.e., a 2D space of uniform negative curvature [30]. Hyperbolic lattices have recently become a fertile ground to investigate interplay of the negative curvature with a variety of physics phenomena, including the Bloch theorem [31][32][33], Hofstadter spectra [34][35][36], topological phases [37][38][39][40][41], and strong correlations [42][43][44][45]. Notably, in the cQED experiment [27], the nearest-neighbor tightbinding model on the so-called heptagon-kagome lattice (Fig.…”

Band theory and boundary modes of high-dimensional representations of infinite hyperbolic lattices

“…Even if a bulk topological index only involves 1D representations (e.g., Ref. 11 ), the corresponding edge states (if exists) must involve higher dimensional non-unitary irreps, because 1D (unitary and non-unitary) representations cannot form edge modes. This observation is one example demonstrating the incompleteness of 1D representations in non-Euclidean lattices.…”

“…The abelian nature of the translation groups in crystals limits their representations to one-dimensional (1D), i.e., the Bloch factor, e ikr , greatly simplifying the mathematical description of waves in crystals. New materials and structures with complex spatial order beyond periodic lattices are being discovered, with a particularly interesting class being hyperbolic lattices, which have recently evolved from pure mathematical concepts 1 to real materials realizable in the lab [2][3][4][5][6][7][8][9][10][11][12] . These lattices are perfectly ordered in hyperbolic space, i.e., space with constant negative curvature.…”

“…How to describe waves in hyperbolic lattices? In recent studies, a range of intriguing features have been reported, e.g., topological edge states 5,11 , higher-genus torus Brillouin zones (BZs) 2,3,6,9 , and circuit quantum electrodynamics 4,12 , outlining an exciting arena of new theories. However, key questions still remain on fundamental principles of constructing wave eigenstates from the symmetries of these hyperbolic lattices.…”