Qunatum mechanics on a torus: Energy levels, Zeeman splitting and transport properties*

Carrier confinement and transport in novel geometries have generated immense theoretical and experimental interest in recent years. Here we investigate the properties of an electron constrained to move on the surface of a torus. We derive the Schrodinger-Riccati equation on a torus using differential geometry. We construct the corresponding action and obtain the eigensolutions through a generalized varaional approach. We show that the dual circular symmetries related to the minor and major radii give rise to at most a 4-fold degeneracy. We obtain the level splitting pattern in the presence of a perpendicular magnetic field. We investigate the transport properties for an electron on a torus with two contacts attached. The torus being a multiply connected topology shows the Ahronov-Bohm effect in presence of a magnetic vector potential and manifests the interference occurring due to electron waves travelling through two different paths. Semiconductor quantum dots and carbon nanotori, for certain chirality, have bandgaps at K-points in the Brillouin zone. Hence they obey nonrelativistic equations, making our study relevant for experimentally amenable outcomes.