Generalizations of Geometric Phase: Fiber Bundles

Arno Böhm, Ali Mostafazadeh, Hiroyasu Koizumi +2 more

Herbert Joseph Bernstein, Anthony V. Phillips

Armin Uhlmann

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

Jun Ohkubo

We propose an application of fiber bundles to counting statistics. The framework of the fiber bundles gives a splitting of a cumulant generating function for a current in a stochastic process, i.e., contributions from the dynamical phase and the geometric phase. We will show that the introduced noncyclic geometric phase is related to kinds of excess contributions, which have been investigated a lot in nonequilibrium physics. Using a specific nonequilibrium model, the characteristics of the noncyclic geometric phase are discussed; in particular, we reveal differences between a geometric contribution for the entropy production and the 'excess entropy production' which has been used to discuss the second law of steady-state thermodynamics.

David S. Simon