Minimal multi-vortex configurations on a flat torus and cylinder

The physics of a planar chiral $p\pm i p$ superconductor is studied for various vortex configurations. The occurrence of vortex quasi-particle bound states is exposed together with their ensuing collective properties, such as sub-gap bands induced by inter-vortex tunneling. A general method to diagonalize the Hamiltonian of a superconductor in the presence of a vortex lattice is developed, that employs only smooth gauge transformations. It renders the Hamiltonian to be periodic (thus allowing the use of a Bloch theorem) and enables the treatment of systems with vortices of finite radii. The pertinent anomalous charge response $c_{xy}$ is calculated (using the Streda formula), and reveals that it contains a quantized contribution. This is attributed to the response to the nucleation of vortices, from which we deduce the system's quantum phase.