hofstadter butterfly

We obtain and analyze Hofstadter's butterfly for the bulk and the edges of the second-order topological insulator (SOTI) subject to an external magnetic field. The applied gauge field is accounted for by performing a Peierls substitution t → texp(i A · dr), modifying the motion of the electrons on the lattice. We compare and contrast the lowest Landau level energy predicted by the continuum surface theory (E = ∆ surf) to the one obtained through numerical simulation. We find that as the applied magnetic flux increases, or equivalently, as the magnetic length decreases to the order of lattice lengths, the magnitude of these Landau levels approaches 0. This is confirmed by a clear signature within the surface gap of Hofstadter's butterfly for the hinge modes: the lowest Landau levels which lie on the sides of the cone emerging from the addition of all possible kz start from E = ∆ surf for Φ = 0 and linearly move towards E = 0, eventually joining the bulk of the sample. This strongly suggests that the physics underlying Hofstadter's butterfly are indeed responsible for the continuum-to-lattice discrepancy.

The self-similar energy spectrum of a particle in a periodic potential under a magnetic field, known as the Hofstadter butterfly, is determined by the lattice geometry as well as the external field. Recent realizations of artificial gauge fields and adjustable optical lattices in cold atom experiments necessitate the consideration of these self-similar spectra for the most general two-dimensional lattice. In a previous work, we investigated the evolution of the spectrum for an experimentally realized lattice which was tuned by changing the unit cell structure but keeping the square Bravais lattice fixed. We now consider all possible Bravais lattices in two dimensions and investigate the structure of the Hofstadter butterfly as the lattice is deformed between lattices with different point symmetry groups. We model the optical lattice by a sinusoidal real space potential and obtain the tight binding model for any lattice geometry by calculating the Wannier functions. We introduce the magnetic field via Peierls substitution and numerically calculate the energy spectrum. The transition between the two most symmetric lattices, i.e. the triangular and the square lattice displays the importance of bipartite symmetry featuring deformation as well as closing of some of the major energy gaps. The transition from the square to rectangular and from the triangular to centered rectangular lattices are analyzed in terms of coupling of one-dimensional chains. We calculate the Chern numbers of the major gaps and Chern number transfer between bands during the transitions. We use gap Chern numbers to identify distinct topological regions in the space of Bravais lattices.