Geometric/Berry Phase Explained | Elementary References

[mtak0114]

Hi
could someone please explain what Geometric/berry phase is I've had a look and there seems to be several ways to interpret the physics. My understanding is that it occurs when your quantum state traces out a closed path in some parameter space, which is some how related to degeneracies in a hamiltonian. What is this parameter space if we are thinking of an electrons spin say? Is it always the same or does it depend specifically on the hamiltonian

thanks in advance

ohhh and if anyone has any good elementary references I would be very interested

Marks

 

[arkajad]

You may like to consult http://books.google.fr/books?id=Yka...page&q=berry phase classical quantum&f=false", which should give you some idea.

 

[mtak0114]

thanks for the references, I had a look at the book but there is still something I'm confused about, the parameter space which people refer to seem to refer to either those parameters which the hamiltonian depend on or the projective hilbert space - the bloch sphere for say a spin half particle. A common example then seems to be an electron in a magnetic field, the hamiltonian is:

$$H = \mu \underline{B}\cdot \underline{\sigma}$$

If $$\underline{B}$$ is the magnetic field directions and the parameters which people refer to. Now if this forms a closed loop in $$\underline{B}$$-space this should result in a geometric phase but for this example there is no difference if I look at the $$\underline{B}$$-space or the Bloch sphere so my question is should the physics be seen (always) as resulting from a closed loop on the bloch sphere?

 

[arkajad]

The example with the Bloch sphere is just an example for spin 1/2 = the simplest non-trivial quantum system, with 2-dimensional space of quantum states. Yet this example contains all the essential features.

 

[mtak0114]

Yes
but is this the correct picture? What I mean is regardless of the hamiltonian if the state forms a closed loop on the bloch sphere will there still be a geometric phase?

thanks again

Mark

 

[arkajad]

Yes. Try reading as much as you can from this short but important paper by Barry Simon:
http://www.physics.princeton.edu/~mcdonald/examples/QM/simon_prl_51_2167_83.pdf"
The paper may use some mathematical terms that are rather advanced, nevertheless you should get some idea about what is going on geometrically with Berry's phase.