Theory and calculation of abelian and non-abelian geometric phase factors with SpinDynamica

Abstract

Cyclic quantum evolution is accompanied by a systematic change in the phase of the initial state vector. This change only depends upon the path traced out by the system itself. Such effects are collectively known as geometric phase factors. The geometric foundations of these phase factors are most elegantly formulated in terms of fibre bundle theory and differential forms, both of which can represent a significant hurdle to master. We present a derivation of the abelian and non-abelian Berry phase in terms of embedded manifolds of linear vector spaces. Embedded manifolds offer the advantage of being less abstract than fibre bundles, and are well-suited for explicit calculations. Essential features of the derivation reduce to matrix–vector manipulations. We further discuss a numerical strategy for the calculation of abelian and non-abelian phase factors. Our approach is based upon Hungarian method and the polar decomposition, and is made freely available as a SpinDynamica addon. Additionally, all derivations and analytic calculations are supported by Mathematica notebooks.