Evolution of the Hofstadter butterfly in a tunable optical lattice

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.

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• Extended Creutz ladder with spin-orbit coupling: a one-dimensional analog of the Kane-Mele model
• Hofstadter Butterfly Evolution in the Space of Two-Dimensional Bravais Lattices
• Driven Hofstadter Butterflies and Related Topological Invariants
• Wannier Diagram and Brown-Zak Fermions of Graphene on Hexagonal Boron-Nitride
• Superfluid transition in the attractive Hofstadter-Hubbard model
• Finite size effects and Hofstadter butterfly in a bosonic Mott insulator with relativistic dispersion background
• Emergent topological properties in spatially modulated sub-wavelength barrier lattices