# Book on Berry phase and its relation to topology

I am searching for a book covering the Berry phase. Griffith has a good outline, but I would like a bit more detail, especially on the relation to topology.

According to this post Ballentine also has a description. Altland and Simons seem to have a description relating to topology, so I'm leaning towards that one, but I haven't managed to have a look at it yet. Anything I have missed or recommendations on which of these is useful?

Although you might find most applications of the concept of Berry's phases in condensed matter physics, it is really present in most areas of physics as it captures the deep connection between geometry and physics. Plainly stated, this connection stems from the fact that for the really interesting problems in physics, wave functions are not functions on the configuration space but rather sections of line bundles associated to principal $U(1)$ bundles. Berry phases are just the holonomies of paths on the principal bundles. (Of course there are many generalizations of this statement to the non-Abelian cases and beyond).

Shapere and Wilczek wrote the in the introduction of their 1989 article collection book: Geometric phases in Physics:

We believe that the concept of a geometric phase, repeating the history of the group concept, will eventually find so may realizations and applications in physics that it will repay study for its own sake, and become part of the lingua franca.

I think that it is not hard to say that their prediction became a reality nowadays and the Berry phase concept has found applications in such a wide variety of phenomena, from topological states of matter to inertial navigation systems and from molecular dynamics to quantum computation and much more.

Since this area is developing rather quickly, I can first refer you to a relatively recent book by Chrus'cin'ski and Jamiolkowski Geometric phases in Classical and quantum mechanics. This book covers many applications of the Berry's phase and has a rather detailed description of its geometric origin.

If you are especially interested in the geometric origin of the Berry's phase, then you can find more advanced material in: The geomeric phase in quantum systems by: Bohm, Mostafazadeh, Koizumi, Niu and Zwanziger.

There are still many open questions related to the geometric description of the geometric phases for open systems. These books are not advanced enough to cover this subject. For this subject I would recommend you to read the recent articles by Viennot and Heydari representing two different approaches on the subject.