Non-Abelian Hyperbolic Band Theory from Supercells.

Area of Science:

• Solid-state physics
• Quantum mechanics
• Materials science

Background:

• Bloch band theory commonly describes wave functions on periodic lattices using Abelian Bloch states.
• Hyperbolic lattices, however, also host non-Abelian Bloch states that have been difficult to analyze.
• Existing analytical methods are insufficient for characterizing these complex states.

Purpose of the Study:

• To develop a systematic method for constructing non-Abelian Bloch states in hyperbolic lattices.
• To enable analytical treatments and efficient computation of their properties.
• To advance the band-theoretic characterization of hyperbolic lattice systems.

Main Methods:

• Adapted solid-state physics concepts of supercells and zone folding.
• Applied Abelian band theory to recursively constructed sequences of supercells.
• Developed a rapidly convergent computational approach for bulk spectra and eigenstates.

Main Results:

• Successfully devised a systematic method for constructing non-Abelian Bloch states.
• Enabled efficient computation of bulk spectra and eigenstates for tight-binding models (both gapless and gapped).
• Demonstrated the method's ability to approximate the thermodynamic limit.

Conclusions:

• The supercell method provides an efficient pathway to analyze non-Abelian Bloch states in hyperbolic lattices.
• This work marks a significant advancement toward a complete band-theoretic understanding of these systems.
• The developed method is crucial for future research in hyperbolic materials and quantum systems.