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

TL;DR

This paper extends band theory to hyperbolic lattices under magnetic fields, revealing Dirac cones and fractal spectra, with potential experimental realization in circuit QED systems.

Contribution

It introduces the first hyperbolic band theory under magnetic fields, demonstrating Dirac cones and fractal spectra on higher genus surfaces.

Findings

Dirac cones form on hyperbolic lattices

Unusual fractal energy spectra observed

First explicit example of massless Dirac states on higher genus surfaces

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.