✎ Geometric Phase [The Physics Travel Guide]

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====== Berry Phase ======

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.

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* A nice laymen explanation of Berry phases is [[https://gravityandlevity.wordpress.com/2015/09/10/guest-post-if-you-walk-in-a-closed-loop-do-you-end-up-where-you-started/|If You Walk in A Closed Loop, Do You End Up Where You Started?]] by Anshul Kogar
  * Another great non-technical introduction is "[[https://michaelberryphysics.files.wordpress.com/2013/07/berry178.pdf|The Geometric Phase]]" by Michael Berry. 
  * https://courses.cit.cornell.edu/ece5390/7_berry_phase.pdf
  
<tabbox Concrete> 
Before Berry reinterpreted geometric phases it was already known that if the parameters of quantum system change slowly the state of the system stays the same but it picks up a phase. This is known as adiabatic theorem.

Berry suggested that the parameters should be varied in such a way that they end up at the values they started with. The phase that a system picks up after performing such a loop in parameter space is now called Berry's phase.

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* For a nice introduction, see [[http://inspirehep.net/record/284501/|Quantum Phases And Angles]] by R. Jackiw
  * A nice textbook that discusses Berry's phase is Griffith "Introduction to Quantum Mechanics", especially chapter 10.
  * Geometric phases also exist in classical mechanics, like for example Hannay's angle. 
For a discussion of Hannay's angle, see Section 4.6.3 [[http://www.damtp.cam.ac.uk/user/tong/dynamics/clas.pdf|here]] and also [[http://cmt.nbi.ku.dk/student_projects/bachelor_theses/BachelorThesisMortenIbMunk-Nielsen.pdf|Geometric phases in classical mechanics]] by  Morten Ib Munk-Nielsen

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

Berry phases are, in some sense, the common origin of all [[advanced_notions:quantum_field_theory:anomalies|anomaly effects]].

This is stated, for example in [[http://www.sciencedirect.com/science/article/pii/0550321388902556?via%3Dihub|Berry's phase, commutators, and the Dirac sea]] by Michael Stone and William E. Goff

</tabbox>

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