Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hyperbolic Band Theory under Magnetic Field and Dirac Cones on a Higher Genus Surface

Kazuki Ikeda, Shoto Aoki, Yoshiyuki Matsuki

TL;DR

This work constructs directly the hyperbolic magnetic Bloch states and finds that they form Dirac cones on a coordinate neighborhood, which is the first explicit example of a massless Dirac state on a higher genus surface.

Abstract

We explore the hyperbolic band theory under a magnetic field for the first time. Our theory is a general extension of the conventional band theory defined on a Euclidean lattice into the band theory on a general hyperbolic lattice/Riemann surface. Our methods and results can be confirmed experimentally by circuit quantum electrodynamics (cQED), which enables us to create novel materials in a hyperbolic space. To investigate the band structures, we construct directly the hyperbolic magnetic Bloch states and find that they form Dirac cones on a coordinate neighborhood, by which they can be regarded as a global quantum gravity solution detectable in a laboratory. Besides this is the first explicit example of a massless Dirac state on a higher genus surface. Moreover we show that the energy spectrum exhibits an unusual fractal structure refracting the negative curvature, when plotted as a function of a magnetic flux.

Hyperbolic Band Theory under Magnetic Field and Dirac Cones on a Higher Genus Surface

TL;DR

This work constructs directly the hyperbolic magnetic Bloch states and finds that they form Dirac cones on a coordinate neighborhood, which is the first explicit example of a massless Dirac state on a higher genus surface.

Abstract

We explore the hyperbolic band theory under a magnetic field for the first time. Our theory is a general extension of the conventional band theory defined on a Euclidean lattice into the band theory on a general hyperbolic lattice/Riemann surface. Our methods and results can be confirmed experimentally by circuit quantum electrodynamics (cQED), which enables us to create novel materials in a hyperbolic space. To investigate the band structures, we construct directly the hyperbolic magnetic Bloch states and find that they form Dirac cones on a coordinate neighborhood, by which they can be regarded as a global quantum gravity solution detectable in a laboratory. Besides this is the first explicit example of a massless Dirac state on a higher genus surface. Moreover we show that the energy spectrum exhibits an unusual fractal structure refracting the negative curvature, when plotted as a function of a magnetic flux.

Paper Structure

This paper contains 6 sections, 9 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Hamiltonian on Hyperbolic Surface under Magnetic Field
  3. Band Structure and Quantum Gravity
  4. Hofstadter Problem on Hyperbolic Tiling
  5. Remarks on the Hall Conductance and the Brillouin Zone
  6. Conclusion and Discussion

Figures (3)

  • Figure 1: The regular tessellation $\{8,8\}$ of the hyperbolic plane, which gives the genus-2 surface.
  • Figure 2: Band structure of the electrons on the $\{8,8\}$ hyperbolic tiling. Unspecified wave numbers are set to 0. The figures correspond to some lower energy bands at $B=1/8$.
  • Figure 3: Spectrum of an electron on a hyperbolic surface $(g=2)$ under a uniform magnetic field, plotted as a function of a rational flux. The plot is generated with the lattice Hamiltonian \ref{['eq:Ham_latt']}.