TY - JOUR
T1 - Topological Hyperbolic Lattices
AU - Yu, Sunkyu
AU - Piao, Xianji
AU - Park, Namkyoo
N1 - Publisher Copyright:
© 2020 American Physical Society.
PY - 2020/7/31
Y1 - 2020/7/31
N2 - Non-Euclidean geometry, discovered by negating Euclid's parallel postulate, has been of considerable interest in mathematics and related fields for the description of geographical coordinates, Internet infrastructures, and the general theory of relativity. Notably, an infinite number of regular tessellations in hyperbolic geometry-hyperbolic lattices-A re expected to extend Euclidean Bravais lattices and the consequent wave phenomena to non-Euclidean geometry. However, topological states of matter in hyperbolic lattices have yet to be reported. Here we investigate topological phenomena in hyperbolic geometry, exploring how the quantized curvature and edge dominance of the geometry affect topological phases. We report a recipe for the construction of a Euclidean photonic platform that inherits the topological band properties of a hyperbolic lattice under a uniform, pseudospin-dependent magnetic field, realizing a non-Euclidean analog of the quantum spin Hall effect. For hyperbolic lattices with different quantized curvatures, we examine the topological protection of helical edge states and generalize Hofstadter's butterfly, by employing two empirical parameters that measure the edge confinement and defect immunity. We demonstrate that the proposed platforms exhibit the unique spectral-magnetic sensitivity of topological immunity in highly curved hyperbolic planes. Our approach is applicable to general non-Euclidean geometry and enables the exploitation of infinite lattice degrees of freedom for band theory.
AB - Non-Euclidean geometry, discovered by negating Euclid's parallel postulate, has been of considerable interest in mathematics and related fields for the description of geographical coordinates, Internet infrastructures, and the general theory of relativity. Notably, an infinite number of regular tessellations in hyperbolic geometry-hyperbolic lattices-A re expected to extend Euclidean Bravais lattices and the consequent wave phenomena to non-Euclidean geometry. However, topological states of matter in hyperbolic lattices have yet to be reported. Here we investigate topological phenomena in hyperbolic geometry, exploring how the quantized curvature and edge dominance of the geometry affect topological phases. We report a recipe for the construction of a Euclidean photonic platform that inherits the topological band properties of a hyperbolic lattice under a uniform, pseudospin-dependent magnetic field, realizing a non-Euclidean analog of the quantum spin Hall effect. For hyperbolic lattices with different quantized curvatures, we examine the topological protection of helical edge states and generalize Hofstadter's butterfly, by employing two empirical parameters that measure the edge confinement and defect immunity. We demonstrate that the proposed platforms exhibit the unique spectral-magnetic sensitivity of topological immunity in highly curved hyperbolic planes. Our approach is applicable to general non-Euclidean geometry and enables the exploitation of infinite lattice degrees of freedom for band theory.
UR - http://www.scopus.com/inward/record.url?scp=85089503427&partnerID=8YFLogxK
U2 - 10.1103/PhysRevLett.125.053901
DO - 10.1103/PhysRevLett.125.053901
M3 - Article
C2 - 32794858
AN - SCOPUS:85089503427
SN - 0031-9007
VL - 125
JO - Physical Review Letters
JF - Physical Review Letters
IS - 5
M1 - 053901
ER -