Hyperbolic band theory under magnetic field and Dirac cones on a higher genus surface

Ikeda, Kazuki; Aoki, Shoto; Matsuki, Yoshiyuki

doi:10.1088/1361-648x/ac24c4

Cited by 37 publications

(17 citation statements)

References 52 publications

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“…Recently, Maciejko and Rayan proposed the first hyperbolic band theory based on the idea of algebraic geometry [62]. Subsequently, hyperbolic lattice band theory under a magnetic field [63] and the crystallography of hyperbolic lattice [64] have further been developed. Thus, research of topological states of matter in hyperbolic lattices based on periodic boundary conditions and hyperbolic topological band theory will be worth investigating in future works.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…Notably, recent experimental realization of hyperbolic lattices in circuit quantum electrodynamics has been reported [60]. This experimental breakthrough inspired several subsequent theoretical studies in hyperbolic lattices [53,54,[61][62][63][64]. In our work, motivated by the recent experimental realization of hyperbolic lattices and theoretical progress in topological hyperbolic lattices, we aim to extend CI phases and consequent topological phenomena to non-Euclidean hyperbolic geometry.…”

Section: Introductionmentioning

confidence: 95%

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Chern insulator in a hyperbolic lattice

Liu,

Hua,

Peng

et al. 2022

Preprint

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Motivated by the recent experimental realizations of hyperbolic lattices in circuit quantum electrodynamics and the research interest in the non-Euclidean generalization of topological phenomena, we investigate the Chern insulator phases in a hyperbolic {8, 3} lattice, which is made from regular octagons (8-gons) such that the coordination number of each lattice site is 3. Based on the conformal projection of the hyperbolic lattice into the Euclidean plane, i.e., the Poincaré disk model, by calculating the Bott index (B) and the two-terminal conductance, we reveal two Chern insulator phases (with B = 1 and B = −1, respectively) accompanied with quantized conductance plateaus in the hyperbolic {8, 3} lattice. The numerical calculation results of the nonequilibrium local current distribution further confirm that the quantized conductance plateau originates from the chiral edge states and the two Chern insulator phases exhibit opposite chirality. Moreover, we explore the effect of disorder on topological phases in the hyperbolic lattice. It is demonstrated that the chiral edge states of Chern insulators are robust against weak disorder in the hyperbolic lattice. More fascinating is the discovery of disorder-induced topological non-trivial phases exhibiting chiral edge states in the hyperbolic lattice, realizing a non-Euclidean analog of topological Anderson insulator. Our work provides a route for the exploration of topological non-trivial states in hyperbolic geometric systems.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

Chern insulator in a hyperbolic lattice

Liu,

Hua,

Peng

et al. 2022

Preprint

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“…Fractional quantum Hall states. With the basic framework of hyperbolic band theory established, the logical next step is to apply a magnetic field [64,37,1]. Our techniques should carry over without much trouble to integer quantum Hall states, where the magnetic flux through a unit cell is an integer multiple N B of the hyperbolic area.…”

Section: Speculationsmentioning

confidence: 99%

“…A subsequent experiment realizes a hyperbolic drum as an electric circuit [44], culminating in the experimental detection of negative curvature via the detection of the eigenmodes of the Laplace-Beltrami operator and of signal propagation along hyperbolic geodesics. Further theoretical works emerging recently in hyperbolic matter and band theory include [37,1,8,64,4,6].…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic band theory through Higgs bundles

Kienzle,

Rayan

2022

Preprint

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Hyperbolic lattices underlie a new form of quantum matter with potential applications to quantum computing and simulation and which, to date, have been engineered artificially. A corresponding hyperbolic band theory has emerged, extending 2-dimensional Euclidean band theory in a natural way to higher-genus configuration spaces. Attempts to develop the hyperbolic analogue of Bloch's theorem have revealed an intrinsic role for algebro-geometric moduli spaces, notably those of stable bundles on a curve. We expand this picture to include Higgs bundles, which enjoy natural interpretations in the context of band theory. First, their spectral data encodes a crystal lattice and momentum, providing a framework for symmetric hyperbolic crystals. Second, they act as a complex analogue of crystal momentum. As an application, we elicit a new perspective on Euclidean band theory. Finally, we speculate on potential interactions of hyperbolic band theory, facilitated by Higgs bundles, with other themes in mathematics and physics.

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“…In particular, hyperbolic band theory (HBT) [5] describes the contribution of Bloch waves to energy bands on two-dimensional (2D) hyperbolic lattices, predicting that momentum space is at least four-dimensional (4D). Among the plethora of recent theoretical studies of physical phenomena on hyperbolic lattices are the continuum approximation of the hyperbolic plane [6], the role of magnetic fields [7][8][9], hyperbolic crystallography [10], automorphic wave functions [11], photon bound states [12], and flat band properties [13,14]. However, the implications of HBT for the band topology in momentum space, and the interplay between topological invariants in position and momentum space on hyperbolic lattices have not been explored so far.…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic Topological Band Insulators

Urwyler¹,

Lenggenhager²,

Boettcher³

et al. 2022

Preprint

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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.

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Hyperbolic band theory under magnetic field and Dirac cones on a higher genus surface

Cited by 37 publications

References 52 publications

Chern insulator in a hyperbolic lattice

Chern insulator in a hyperbolic lattice

Hyperbolic band theory through Higgs bundles

Hyperbolic Topological Band Insulators

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