Evolution of the Hofstadter butterfly in a tunable optical lattice

TL;DR

This paper explores how the Hofstadter butterfly energy spectrum evolves in a tunable optical lattice, revealing topological transitions and the merging of spectral features as lattice geometry changes, with implications for experimental detection of topological phases.

Contribution

It provides a detailed calculation of Hofstadter butterflies in an adjustable lattice, showing topologically non-trivial merging of spectra and Chern number transfer during geometric evolution.

Findings

Three distinct regimes of Hofstadter butterflies identified.

Merging of square lattice butterflies into a honeycomb lattice butterfly.

Topological gap closings and Chern number transfers observed.

Abstract

Recent advances in realizing artificial gauge fields on optical lattices promise experimental detection of topologically non-trivial energy spectra. Self-similar fractal energy structures generally known as Hofstadter butterflies depend sensitively on the geometry of the underlying lattice, as well as the applied magnetic field. The recent demonstration of an adjustable lattice geometry [L. Tarruell \textit{et al.}, Nature 483, 302--305 (2012)] presents a unique opportunity to study this dependence. In this paper, we calculate the Hofstadter butterflies that can be obtained in such an adjustable lattice and find three qualitatively different regimes. We show that the existence of Dirac points at zero magnetic field does not imply the topological equivalence of spectra at finite field. As the real-space structure evolves from the checkerboard lattice to the honeycomb lattice, two square lattice Hofstadter butterflies merge to form a honeycomb lattice butterfly. This merging is topologically non-trivial, as it is accomplished by sequential closings of gaps. Ensuing Chern number transfer between the bands can be probed with the adjustable lattice experiments. We also calculate the Chern numbers of the gaps for qualitatively different spectra and discuss the evolution of topological properties with underlying lattice geometry.