Syed Ali Raza Supervisor: Dr. Pervez Hoodbhoy. Topological Insulators. What are Topological insulators?. Fairly recently discovered electronic phases of matter. Theoretically predicted in 2005 and 2007 by Zhang, Zahid Hassan and Moore. Experimentally proven in 2007.
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If the particle was initially in the nth state of Hi then it will be carried to the nth state of Hf .
“A system slowly transported round a circuit will return to it’s original state; this is the content of the adiabatic theorem. Moreover it’s internal clocks will register the passage of time; this can be regarded as the dynamical phase factor. The remarkable and rather mysterious result of this paper is in addition the system records its history in a deeply geometrical way.” – Berry 1984
In a polar coordinates parameter space we define a spinor for the hamiltonian
A global vector potential is not possible in the presence of a magnetic monopole.
There is a singularity which is equal to the full monopole flux
The vector potential does not describe a monopole at the origin, but one where a tiny tube (the dirac string) comes up the negative z axis, smuggling in the entire flux.
As it is spherically symmetric, we can move the dirac string any where on the sphere with a gauge transformation.
Patch up the two different vector potentials at the equator.
The two potentials differ by a single valued gauge transformation and you can recover the diracquantisationcondition from it.
You can also get this result by Holonomy (Wilczek)