Spectral Flow and Global Topology of the Hofstadter Butterfly

We study the relation between the global topology of the Hofstadter butterfly of a multiband insulator and the topological invariants of the underlying Hamiltonian. The global topology of the butterfly, i.e., the displacement of the energy gaps as the magnetic field is varied by one flux quantum, is determined by the spectral flow of energy eigenstates crossing gaps as the field is tuned. We find that for each gap this spectral flow is equal to the topological invariant of the gap, i.e., the net number of edge modes traversing the gap. For periodically driven systems, our results apply to the spectrum of quasienergies. In this case, the spectral flow of the sum of all the quasienergies gives directly the Rudner-Lindner-Berg-Levin invariant that characterizes the topological phases of a periodically driven system.

Figure 1

Examples of Hofstadter butterfly spectra for (a) a simple charged particle on an infinite square lattice (Harper model), (b) a topologically trivial multiband insulator (Qi-Wu-Zhang model with $u=3$ of Ref. [16]), (c) a topological multiband insulator (same Qi-Wu-Zhang model with $u=-1.2$), and (d) a topological Floquet insulator [Rudner model of Ref. [17] with ${\delta }_{AB}=0$ and $JT/5$ chosen for the first four segments as $(3/8,3/8,5/8,5/8)\pi$]. The topological indices of a few representative band gaps are shown in the figure.

Figure 2

Examples of Hofstadter spectra analogous to those shown in Fig. 1, but computed for a lattice of finite width along $y$ (24 sites with periodic boundary conditions), and still infinite along $x$. For better visibility of the spectral flow, the spectra are shown for a given quasimomentum only ($k_x=0$).

Figure 3

Spectra of the Qi-Wu-Zhang model as a function of the magnetic flux (two cycles shown), computed on a lattice of finite width (36 sites) along $y$ and for a fixed quasimomentum ($k_x=0$). $\gamma$ indicates the hopping amplitude at the edge ($\gamma =0$ for open and $\gamma =1$ for periodic boundary conditions). States with more than 30% weight in the first two rows at the top (bottom) edge are marked by thick red (blue) lines.