Non-Abelian hyperbolic band theory from supercells

Abstract

Wave functions on periodic lattices are commonly described by Bloch band theory. Besides Abelian Bloch states labeled by a momentum vector, hyperbolic lattices support non-Abelian Bloch states that have so far eluded analytical treatments. By adapting the solid-state-physics notions of supercells and zone folding, we devise a method for the systematic construction of non-Abelian Bloch states. The method applies Abelian band theory to sequences of supercells, recursively built as symmetric aggregates of smaller cells, and enables a rapidly convergent computation of bulk spectra and eigenstates for both gapless and gapped tight-binding models. Our supercell method provides an efficient means of approximating the thermodynamic limit and marks a pivotal step towards a complete band-theoretic characterization of hyperbolic lattices.

5 pages (5 figures) + references + 19 pages of Supplemental Material (10 figures + 2 tables); (v2) updated with accepted version

References in corpus (5)

• Hyperbolic band topology with non-trivial second Chern numbers
• Periodic boundary conditions on the pseudosphere
• Density of states of tight-binding models in the hyperbolic plane
• Symmetry and topology of hyperbolic Haldane models
• Converging Periodic Boundary Conditions and Detection of Topological Gaps on Regular Hyperbolic Tessellations

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• Tessellation codes: encoded quantum gates by geometric rotation
• Discrete holography and density of states in the crossover from hyperbolic to Euclidean lattices
• Hofstadter's Butterfly in AdS${}_3$ Black Holes
• Experimental Proposal on Non-Abelian Aharonov-Bohm Caging Effect with a Single Trapped Ion