The quantum phase 2-form near degeneracies: two numerical studies

Abstract

The phase of a quantum state changes rapidly as parameters X = ( X 1 , X 2 ,...) are varied near a degeneracy X *, reflecting the monopole singularity of the underlying phase 2-form V ( X ) at X *. The singularities may be sources or sinks of V . We study them numerically and display them graphically for two families of hamiltonians whose degeneracy structure is typical. First is a particle moving along a line segment with kinetic energy quartic in the momentum (‘quartic-momentum square well’); the X are incorporated into the boundary conditions. Second is a charged particle moving in a domain D of the plane which is threaded by a magnetic flux line of strength α, with wavefunction vanishing on the boundary ∂D (‘Aharonov-Bohm billiards’); the X are α and parameters specifying ∂D; V is not invariant under gauge transformations of the vector potential generating the flux. For Aharonov-Bohm billiards we study how the spatial patterns of phase of wavefunctions change round circuits near degeneracies; these patterns also have singularities (wavefront dislocations) that appear and disappear by colliding with each other and with ∂D.