Selberg trace formula in hyperbolic band theory
Adil Attar, Igor Boettcher
Abstract
We apply Selberg's trace formula to solve problems in hyperbolic band theory, a recently developed extension of Bloch theory to model band structures on experimentally realized hyperbolic lattices. For this purpose we incorporate the higher-dimensional crystal momentum into the trace formula and evaluate the summation for periodic orbits on the Bolza surface of genus two. We apply the technique to compute partition functions on the Bolza surface and propose an approximate relation between the lowest bands on the Bolza surface and on the {8,3} hyperbolic lattice. We discuss the role of automorphism symmetry and its manifestation in the trace formula.
Topics & Concepts
Selberg trace formulaTRACE (psycholinguistics)MathematicsLattice (music)Mathematical analysisPartition (number theory)Extension (predicate logic)Pure mathematicsPhysicsRiemann hypothesisCombinatoricsLinguisticsProgramming languageAcousticsPhilosophyComputer scienceGeometric and Algebraic TopologyQuantum chaos and dynamical systemsAdvanced Combinatorial Mathematics