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Selberg trace formula in hyperbolic band theory.
Adil Attar1, Igor Boettcher1,2
1Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada.
This study introduces hyperbolic band theory using Selberg's trace formula to analyze band structures on hyperbolic lattices. Researchers computed partition functions and proposed relationships between different hyperbolic surfaces.
Area of Science:
- Condensed matter physics
- Mathematical physics
- Non-Euclidean geometry
Background:
- Bloch theory describes electron behavior in crystals.
- Hyperbolic band theory extends Bloch theory to hyperbolic geometry.
- Selberg's trace formula relates spectral data to geometric properties.
Purpose of the Study:
- To apply Selberg's trace formula to hyperbolic band theory.
- To model band structures on hyperbolic lattices.
- To investigate the Bolza surface and {8,3} hyperbolic lattice.
Main Methods:
- Incorporating higher-dimensional crystal momentum into the trace formula.
- Evaluating summations for periodic orbits on the Bolza surface.
- Computing partition functions using the trace formula.
Main Results:
- Successfully applied Selberg's trace formula to hyperbolic band theory problems.
- Computed partition functions on the Bolza surface.
- Proposed an approximate relation between band structures on the Bolza surface and the {8,3} hyperbolic lattice.
Conclusions:
- Selberg's trace formula is a viable tool for hyperbolic band theory.
- Automorphism symmetry plays a significant role in the trace formula for these systems.
- This work bridges concepts from geometry, topology, and condensed matter physics.
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