A Short Introduction to: Connections on $$ \mathbb{U} $$ (1) Bundles and Berry’s Phase.

Arno Böhm, Luis J. Boya, Ali Mostafazadeh +1 more

Igor Evgenievich Pris, E. A. Tolkachev, Yakov Shnir

Juan Andrés Ospina Sabogal

Geometric phases form a rich subject in mathematical physics, lying at the intersection of differential geometry, functional analysis, and topology. This combination of ideas can make the subject difficult to approach for advanced undergraduate students, even though it is very rewarding. The aim of this thesis is to give a self-contained introduction to the mathematical framework behind geometric phases. Assuming only basic differential geometry, we develop the theory of vector bundles, connections, curvature, and holonomy, and use it to describe the Berry and Wilczek–Zee phases as holonomies of natural connections on complex vector bundles. We also discuss the Aharonov–Bohm effect as an example of a flat connection with globally nontrivial holonomy.

Barry Simon A

It is shown that the "geometrical phase factor" recently found by Berry in his study of the quantum adiabatic theorem is precisely the holonomy in a Hermitian line bundle since the adiabatic theorem naturally defines a connection in such a bundle. This not only takes the mystery out of Berry's phase factor and provides calculational simple formulas, but makes a connection between Berry's work and that of Thouless et al. This connection allows the author to use Berry's ideas to interpret the integers of Thouless et al. in terms of eigenvalue degeneracies.

Mikhail Orionovich Katanaev

A geometric interpretation of Berry’s phase, its Wilczek–Zee non-Abelian generalization, and Aharonov–Bohm effect is given in terms of connections on principal fiber bundles. It is shown that a principal fiber bundle in all cases can be trivial while the connection and its holonomy group are nontrivial. Therefore the main role is played not by topological effects but geometrical ones. The Berry phase is also shown not to be connected to the adiabatic changes of the Hamiltonian parameters. 1