Topological Winding Number Change and Broken Inversion Symmetry in a Hofstadter's Butterfly

Graphene's quantum Hall features are associated with a pi Berry's phase due to its odd topological pseudospin winding number. In nearly aligned graphenehexagonal BN heterostructures, the lattice and orientation mismatch produce a superlattice potential, yielding secondary Dirac points in graphene's electronic spectrum, and under a magnetic field, a Hofstadter butterfly-like energy spectrum. Here we report an additional pi Berry's phase shift when tuning the Fermi level past the secondary Dirac points, originating from a change in topological winding number from odd to even when the Fermi-surface electron orbit begins to enclose the secondary Dirac points. At large hole doping inversion symmetry breaking generates a distinct hexagonal pattern in the longitudinal resistivity versus magnetic field and charge density. Major Hofstadter butterfly features persist up to similar to 100 K, demonstrating the robustness of the fractal energy spectrum in these systems.