Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algebra of Hyperbolic Band Theory under Magnetic Field

Kazuki Ikeda, Yoshiyuki Matsuki, Shoto Aoki

Abstract

We explore algebras associated with the hyperbolic band theory under a magnetic field for the first time. We define the magnetic Fuchsian group associated with a higher genus Riemann surface. By imposing the magnetic boundary conditions for the hyperbolic Bloch states, we construct the hyperbolic magnetic Bloch states and investigate their energy spectrum. We give a connection between such magnetic Bloch states and automorphic forms. Our theory is a general extension of the conventional algebra associated with the band theory defined on a Euclidean lattice/space into that of the band theory on a general hyperbolic lattice/Riemann surface.

Algebra of Hyperbolic Band Theory under Magnetic Field

Abstract

We explore algebras associated with the hyperbolic band theory under a magnetic field for the first time. We define the magnetic Fuchsian group associated with a higher genus Riemann surface. By imposing the magnetic boundary conditions for the hyperbolic Bloch states, we construct the hyperbolic magnetic Bloch states and investigate their energy spectrum. We give a connection between such magnetic Bloch states and automorphic forms. Our theory is a general extension of the conventional algebra associated with the band theory defined on a Euclidean lattice/space into that of the band theory on a general hyperbolic lattice/Riemann surface.

Paper Structure

This paper contains 10 sections, 7 theorems, 103 equations, 2 figures.

Table of Contents

  1. Introduction and Summary
  2. Magnetic Field on Hyperbolic Surface
  3. General Setup
  4. General Prescriptions for States in Hyperbolic Surface under Magnetic Field
  5. Magnetic Fuchsian Group
  6. Review on the Fuchsian Group
  7. Magnetic Fuchsian Group
  8. Hyperbolic Bloch States under Magnetic Field
  9. Automorphic Forms
  10. Summary

Key Result

Proposition 3.2

For arbitrary $z_0 \in \mathbb{H}$, an orbit $\Set{e^{\theta S} z_0| \theta \in \mathbb{R} }$ is equal to where Therefore the orbit generated by $S$ is a circle with center $z=ia$ and radius $b$.

Figures (2)

  • Figure 1: $\{8,8\}$-tiling of the Poincar'e disk.
  • Figure 2: A fundamental domain of $\{8,8\}$ tiling.

Theorems & Definitions (16)

  • Definition 3.1: Hyperbolic plane
  • Proposition 3.2
  • Proof
  • Proposition 3.3
  • Definition 3.4
  • Lemma 3.5
  • Proposition 3.6
  • Proof
  • Theorem 3.7
  • Proof
  • ...and 6 more