On the Hyperbolic Bloch Transform

Nagy, Ákos; Rayan, Steven

doi:10.1007/s00023-023-01336-8

Cited by 5 publications

(3 citation statements)

References 16 publications

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“…The form under the integral sign is called the pushback of α by f , and we denote it as α. 14 A standard result in differential geometry [63] states that…”

Section: Matrix-valued Differential Formsmentioning

confidence: 99%

“…Let us comment on the symmetry of the matrix-valued differential forms F and A in band indices. Since U † U = 1 for normalized eigenstates, it follows by taking the exterior 14 Conventionally, the pushback of α by f is labeled as f * α. However, our work reserves the "star" symbol for complex conjugation.…”

Section: Berry-wilczek-zee Connection and Chern Numbersmentioning

confidence: 99%

“…[4][5][6][7][8]. Other recently investigated aspects of hyperbolic lattices include hyperbolic band theory [9,10] and crystallography [11], non-Abelian Bloch states [10,[12][13][14][15], periodic boundary conditions [10,[16][17][18][19]] and the thermodynamic limit [18][19][20], Hofstadter spectra [8,[21][22][23], effects of strong correlations [17,[24][25][26], continuum approximation [27], exact trace formulas [28], elastic vibrations [29], flat bands [1,[30][31][32][33], connections to holography [34][35][36][37][38], and higher-order topological phenomena [39,40]. It is by now established that hyperbolic lattices exhibit single-particle dynamics and many-body phenomena richly distinct from those observed in Euclidean lattices.…”

Section: Introductionmentioning

confidence: 99%

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Symmetry and topology of hyperbolic Haldane models

et al. 2023

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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.

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“…The form under the integral sign is called the pushback of α by f , and we denote it as α. 14 A standard result in differential geometry [63] states that…”

Section: Matrix-valued Differential Formsmentioning

confidence: 99%

Section: Berry-wilczek-zee Connection and Chern Numbersmentioning

confidence: 99%