Spectral and Combinatorial Aspects of Cayley-Crystals

Abstract

Owing to their interesting spectral properties, the synthetic crystals over lattices other than regular Euclidean lattices, such as hyperbolic and fractal ones, have attracted renewed attention, especially from materials and meta-materials research communities. They can be studied under the umbrella of quantum dynamics over Cayley graphs of finitely generated groups. In this work, we investigate numerical aspects related to the quantum dynamics over such Cayley graphs. Using an algebraic formulation of the "periodic boundary condition" due to Lueck [Geom. Funct. Anal. 4, 455-481 (1994)], we devise a practical and converging numerical method that resolves the true bulk spectrum of the Hamiltonians. Exact results on the matrix elements of the resolvent, derived from the combinatorics of the Cayley graphs, give us the means to validate our algorithms and also to obtain new combinatorial statements. Our results open the systematic research of quantum dynamics over Cayley graphs of a very large family of finitely generated groups, which includes the free and Fuchsian groups.

converging periodic bc for hyperbolic and fractal crystals, tested against exact results

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• Observation of novel topological states in hyperbolic lattices
• Automorphic Bloch theorems for hyperbolic lattices
• Universality of Hofstadter butterflies on hyperbolic lattices
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Cited by in corpus (10)

• Non-Abelian hyperbolic band theory from supercells
• Anderson localization transition in disordered hyperbolic lattices
• Hyperbolic non-Abelian semimetal
• Converging Periodic Boundary Conditions and Detection of Topological Gaps on Regular Hyperbolic Tessellations
• Topological linear response of hyperbolic Chern insulators
• Hyperbolic lattices and two-dimensional Yang-Mills theory
• Non-Hermitian Skin Effect Along Hyperbolic Geodesics
• Hyperbolic Spin Liquids
• Classifying the Dynamics of Architected Materials by Groupoid Methods
• A $C^{\ast }$-algebraic view on the interaction of real- and reciprocal space topology in skyrmion crystals