Electronic structure of periodic curved surfaces: Topological band structure

The electronic band structure for electrons bound on periodic minimal surfaces is differential-geometrically formulated and numerically calculated. We focus on minimal surfaces because they are not only mathematically elegant (with the surface characterized completely in terms of "navels") but represent the topology of real systems such as zeolites and negative-curvature fullerenes. The band structure turns out to be primarily determined by the topology of the surface, i.e., how the wave function interferes on a multiply connected surface, so that the bands are little affected by the way in which we confine the electrons on the surface (thin-slab limit or zero thickness from the outset). Another curiosity is that different minimal surfaces connected by the Bonnet transformation (such as Schwarz's P and D surfaces) possess one-to-one correspondence in their band energies at Brillouin-zone boundaries.