Non-Abelian Hyperbolic Band Theory from Supercells

Lenggenhager, Patrick M.; Maciejko, Joseph; Bzdušek, Tomáš

doi:10.1103/physrevlett.131.226401

Cited by 9 publications

(2 citation statements)

References 41 publications

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“…We note that the authors in [314] calculate the bulk DOS in the BBH model on a {6, 4} hyperbolic lattice that preserves the hyperbolic translational symmetry by a supercell method and find that the DOS exhibits a gap closing at a critical point. However, the bulk-edge correspondence has not been explored.…”

Section: Hyperbolic Latticesmentioning

confidence: 93%

Higher-order topological phases in crystalline and non-crystalline systems: a review

Yang,

Wang,

et al. 2024

J. Phys.: Condens. Matter

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In recent years, higher-order topological phases have attracted great interest in various fields of physics. These phases have protected boundary states at lower-dimensional boundaries than the conventional first-order topological phases due to the higher-order bulk-boundary correspondence. In this review, we summarize current research progress on higher-order topological phases in both crystalline and non-crystalline systems. We firstly introduce prototypical models of higher-order topological phases in crystals and their topological characterizations. We then discuss effects of quenched disorder on higher-order topology and demonstrate disorder-induced higher-order topological insulators. We also review the theoretical studies on higher-order topological insulators in amorphous systems without any crystalline symmetry and higher-order topological phases in non-periodic lattices including quasicrystals, hyperbolic lattices, and fractals, which have no crystalline counterparts. We conclude the review by a summary of experimental realizations of higher-order topological phases and discussions on potential directions for future study.

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Section: Hyperbolic Latticesmentioning

confidence: 93%

Higher-order topological phases in crystalline and non-crystalline systems: a review

Yang,

Wang,

et al. 2024

J. Phys.: Condens. Matter

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“…Discrete translational symmetry supported in Euclidean lattices is a cornerstone of modern physics, as it enables the propagation of Bloch waves associated with a commutative translation group in Euclidean spaces. 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 .…”

Section: Introductionmentioning

confidence: 99%

Anomalous and Chern topological waves in hyperbolic networks

Chen,

Zhang,

Qin

et al. 2024

Nat Commun

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Hyperbolic lattices are a new type of synthetic materials based on regular tessellations in non-Euclidean spaces with constant negative curvature. While so far, there has been several theoretical investigations of hyperbolic topological media, experimental work has been limited to time-reversal invariant systems made of coupled discrete resonances, leaving the more interesting case of robust, unidirectional edge wave transport completely unobserved. Here, we report a non-reciprocal hyperbolic network that exhibits both Chern and anomalous chiral edge modes, and implement it on a planar microwave platform. We experimentally evidence the unidirectional character of the topological edge modes by direct field mapping. We demonstrate the topological origin of these hyperbolic chiral edge modes by an explicit topological invariant measurement, performed from external probes. Our work extends the reach of topological wave physics by allowing for backscattering-immune transport in materials with synthetic non-Euclidean behavior.

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Scalable and Programmable Emulation of Photonic Hyperbolic Lattices

Park,

Piao,

2024

ACS Photonics

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Non-Euclidean geometry, which violates the traditional parallel postulate, encompasses a wide class of physical systems, including fullerenes, space-time structures, and complex networks. A notable example is the hyperbolic lattice, which exhibits unique band properties and topological phenomena compared with those of Euclidean lattices. Although hyperbolic lattices have been effectively realized on a Euclidean plane using projections onto the Poincarédisk, the poor scalability of this approach hinders large-size implementations essential for examining lattice physics. Here, we demonstrate a scalable and reconfigurable emulation of wave dynamics in a hyperbolic lattice by employing programmable photonics. We develop a one-dimensional coupledresonator lattice with reconfigurable gauge fields and resonances, which emulates unitary wave evolutions inside a hyperbolic lattice. The programmability of our system allows for modeling time-varying hyperbolic lattices including dynamic defects. The superior scalability of our system compared to the realizations using the Poincarédisk and spatial-domain circuits paves the way to extending non-Euclidean photonics into large-scale and dynamical systems.

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Non-Abelian Hyperbolic Band Theory from Supercells

Cited by 9 publications

References 41 publications

Higher-order topological phases in crystalline and non-crystalline systems: a review

Higher-order topological phases in crystalline and non-crystalline systems: a review

Anomalous and Chern topological waves in hyperbolic networks

Scalable and Programmable Emulation of Photonic Hyperbolic Lattices

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