Gaplessness of Landau Hamiltonians on hyperbolic half-planes via coarse geometry

Abstract

We use coarse index methods to prove that the Landau Hamiltonian on the hyperbolic half-plane, and even on much more general imperfect half-spaces, has no spectral gaps. Thus the edge states of hyperbolic quantum Hall Hamiltonians completely fill up the gaps between Landau levels, just like those of the Euclidean counterpart.

arXiv comment23 pages, 4 figures; Added proofs in Section 1.4; further minor revisions; to appear in Comm. Math. Phys

References in corpus (7)

• Hyperbolic Lattices in Circuit Quantum Electrodynamics
• Quantum Simulation of Hyperbolic Space with Circuit Quantum Electrodynamics: From Graphs to Geometry
• Quantum Hall Effect on the Hyperbolic Plane
• Coarse geometry and topological phases
• Cobordism invariance of topological edge-following states
• Index formulas and charge deficiencies on the Landau levels
• Topological phases on the hyperbolic plane: fractional bulk-boundary correspondence

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• Quantum geometric tensors from sub-bundle geometry
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• A vanishing theorem in $K$-theory for spectral projections of a non-periodic magnetic Schrödinger operator