Intuitive

When a system moves once around a close loop, like for example a pendulum once around its suspension, we expect that it returns to exactly the same state that it started in.

However, this is not always the case. System can pick up a pick up a phase while moving once around a closed loop.

A famous example is a Foucault pendulum. Such a pendulum is expected to return to its original position after a full rotation of the earth in 24 hrs. However, it doesn’t. It picks up an angle, called Hannay’s angle.

• A nice laymen explanation of Berry phases is If You Walk in A Closed Loop, Do You End Up Where You Started? by Anshul Kogar
• https://michaelberryphysics.files.wordpress.com/2013/07/berry178.pdf and
• https://courses.cit.cornell.edu/ece5390/7_berry_phase.pdf

Concrete

• For a nice introduction, see Quantum Phases And Angles by R. Jackiw
• For a discussion of Hannays angle, see Section 4.6.3 here.

Abstract

Why is it interesting?

Berry phases are, in some sense, the common origin of all anomaly effects.

This is stated, for example in Berry's phase, commutators, and the Dirac sea by Michael Stone and William E. Goff