Hofstadter's Butterfly in AdS$_3$ Black Holes

TL;DR

This paper constructs a gauge-covariant lattice model on the BTZ black hole background to study spectral properties, revealing how curvature and horizon size influence quantum fractal structures and responses.

Contribution

It introduces a geometric lattice model on BTZ backgrounds with exact curved Harper equations, linking black hole geometry to quantum spectral phenomena.

Findings

Weaker curvature sharpens butterfly fractal structures.

Larger horizons suppress magnetic and Aharonov--Bohm responses.

Near-horizon states become weakly dispersing with larger horizons.

Abstract

We derive the reduced Dirac Hamiltonian on the non-rotating BTZ background and use its redshift structure to construct a gauge-covariant single-band lattice model on the constant-time BTZ cylinder. In equal-area coordinates the AdS radius $L$ fixes the local Gaussian curvature, while the horizon radius $r_h$ fixes the throat size and the strength of the near-horizon redshift. The lattice model therefore has a direct geometric interpretation and is not presented as an unshown reduction of the two-component Dirac lattice. Its angular Fourier transform yields an exact curved Harper equation with BTZ-dependent hopping amplitudes and a consistent dimensionless angular quasi-momentum. We then supplement global parameter scans with state-resolved diagnostics: spectra color-coded by mean radius, local density of states, direct flux-response versus radius correlations, and Aharonov--Bohm spectral flow and persistent current on the BTZ cycle. These results show that weaker curvature sharpens the butterfly-like fragmentation, whereas larger horizons suppress both magnetic and Aharonov--Bohm response by creating weakly dispersing near-horizon states.