An example of phase holonomy in WKB theory

Abstract

Discusses the application of WKB theory to Harper's equation psi n+1+ psi n-1+2 alpha cos(2 pi beta n+ delta ) psi n=E psi m, in the case in which beta is very close to a rational number, p/q. The WKB wavefunction for this system is a vector valued quantity, proportional to an eigenvector u of a matrix H(x,p), which is parametrised by the phase space coordinates x and p. The complex phase of u is determined by a non-holonomic connection rule; when transported around a cycle and in phase space, u is multiplied by a phase factor ei gamma (c). This phase change manifests itself as a modification of the Bohr-Sommerfeld quantisation conditions.