Chern insulator in a hyperbolic lattice

Liu, Zhengrong; Hua, Chun-Bo; Peng, Tan; Zhou, Bin

doi:10.1103/physrevb.105.245301

Cited by 27 publications

(17 citation statements)

References 123 publications

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“…This work paves the way for several highly exciting future research directions in both experimental and theoretical condensed matter physics. Experimentally, the tunable complex-phase element developed here can be utilized in topolectrical networks to simulate Hamiltonians with topological ground states, such as the recently discovered hyperbolic topological band insulators 28 , 39 or hyperbolic Hofstadter butterfly models 17 , 32 . In particular, local probes in electric circuits provide access to the complete characterization of the Bloch eigenstates, giving the necessary input to compute any topological invariant.…”

Section: Discussionmentioning

confidence: 99%

Hyperbolic matter in electrical circuits with tunable complex phases

et al. 2023

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Curved spaces play a fundamental role in many areas of modern physics, from cosmological length scales to subatomic structures related to quantum information and quantum gravity. In tabletop experiments, negatively curved spaces can be simulated with hyperbolic lattices. Here we introduce and experimentally realize hyperbolic matter as a paradigm for topological states through topolectrical circuit networks relying on a complex-phase circuit element. The experiment is based on hyperbolic band theory that we confirm here in an unprecedented numerical survey of finite hyperbolic lattices. We implement hyperbolic graphene as an example of topologically nontrivial hyperbolic matter. Our work sets the stage to realize more complex forms of hyperbolic matter to challenge our established theories of physics in curved space, while the tunable complex-phase element developed here can be a key ingredient for future experimental simulation of various Hamiltonians with topological ground states.

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Section: Discussionmentioning

confidence: 99%

Hyperbolic matter in electrical circuits with tunable complex phases

et al. 2023

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“…[25], while another model of Chern insulator on the f8; 3g lattice, although lacking translation symmetry, has been considered by Ref. [26]. However, these works did not apply HBT to characterize the models, thus lacking the momentum-space language.…”

Section: Haldanementioning

confidence: 99%

“…The range of recently investigated physical phenomena further includes the effects of magnetic fields [10][11][12], continuum approximation [13], periodic boundary conditions [14][15][16], hyperbolic crystallography [17], photon bound states [18], exact trace formulas [19], Bose-Hubbard model [15], elastic vibrations [20], and flat bands [21][22][23][24]. Notably, two very recent works proposed concrete models of hyperbolic topological insulators [25,26]; however, a systematic investigation of topological quantum numbers on hyperbolic lattices remains largely unexplored.…”

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confidence: 99%

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Hyperbolic Topological Band Insulators

et al. 2022

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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 nontrivial 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 modeling propagation of edge excitations, and by their robustness against disorder.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Bzdušek

Maciejko²

2022

Phys. Rev. B

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Motivated by the recent experimental realizations of hyperbolic lattices in circuit quantum electrodynamics and in classical electric-circuit networks, we study flat bands and band-touching phenomena in such lattices. We analyze noninteracting nearest-neighbor hopping models on hyperbolic analogs of the kagome and dice lattices with heptagonal and octagonal symmetries. We show that two characteristic features of the energy spectrum of those models, namely, the fraction of states in the flat band as well as the number of touching points between the flat band and the dispersive bands, can both be captured exactly by a combination of real-space topology arguments and a reciprocal-space description via the formalism of hyperbolic band theory. Furthermore, using real-space numerical diagonalization on finite lattices with periodic boundary conditions, we obtain insights into higher-dimensional irreducible representations of the non-Euclidean (Fuchsian) translation group of hyperbolic lattices. First, we find that the fraction of states in the flat band is the same for Abelian and non-Abelian hyperbolic Bloch states. Second, we find that only Abelian states participate in the formation of touching points between the flat and dispersive bands.

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Chern insulator in a hyperbolic lattice

Cited by 27 publications

References 123 publications

Hyperbolic matter in electrical circuits with tunable complex phases

Hyperbolic matter in electrical circuits with tunable complex phases

Hyperbolic Topological Band Insulators

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

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