Symmetry and topology of hyperbolic Haldane models

Abstract

Particles hopping on a two-dimensional hyperbolic lattice feature unconventional energy spectra and wave functions that provide a largely uncharted platform for topological phases of matter beyond the Euclidean paradigm. Using real-space topological markers as well as Chern numbers defined in the higher-dimensional momentum space of hyperbolic band theory, we construct and investigate hyperbolic Haldane models, which are generalizations of Haldane's honeycomb-lattice model to various hyperbolic lattices. We present a general framework to characterize point-group symmetries in hyperbolic tight-binding models, and use this framework to constrain the multiple first and second Chern numbers in momentum space. We observe several topological gaps characterized by first Chern numbers of value 1 and 2. The momentum-space Chern numbers respect the predicted symmetry constraints and agree with real-space topological markers, indicating a direct connection to observables such as the number of chiral edge modes. With our large repertoire of models, we further demonstrate that the topology of hyperbolic Haldane models is trivialized for lattices with strong negative curvature.

Anomalous and Chern topological waves in hyperbolic networks

“…In analogy with the Euclidean case, hyperbolic lattices that exhibit discrete periodicity in hyperbolic spaces can also support Bloch waves but with a twist: they belong to a non-commutative group of hyperbolic translations, the Fuchsian group Γ ⊂ PSU(1, 1) 2 – 8 . This complexity therefore results in distinct spectra 9 and band topology in hyperbolic momentum space, as theoretically proposed in the hyperbolic analogues of the quantum spin Hall effect 10 , Chern insulator 11 , higher-order topology 12 , Haldane 13 and Kane–Mele models 5 . Experimental demonstrations, all of which, however, are implemented in reciprocal systems, have been conducted in circuit quantum electrodynamics 14 and topolectrical circuits 15 – 18 .…”

Tuning Second Chern Number in a Four-Dimensional Topological Insulator by High-Frequency Time-Periodic Driving

“…The second Chern number is used to describe the nonlinear response of the current to an electric field and a magnetic field in the 4D TI. [21] It is given by the following formula: [23][24][25]111,112]…”

Anomalous and Chern topological waves in hyperbolic networks

“…In analogy with the Euclidean case, hyperbolic lattices that exhibit discrete periodicity in hyperbolic spaces can also support Bloch waves but with a twist: they belong to a non-commutative group of hyperbolic translations, the Fuchsian group Γ ⊂ P SU (1, 1) [2][3][4][5][6][7][8]. This complexity therefore results in distinct spectra [9] and band topology in hyperbolic momentum space, as theoretically proposed in the hyperbolic analogs of the quantum spin Hall effect [10], Chern insulator [11], higher-order topology [12], Haldane [13] and Kane-Mele models [5]. Experimental demonstrations, all of which, however, are implemented in reciprocal systems, have been conducted in circuit quantum electrodynamics [14] and topolectrical circuits [15][16][17][18].…”