1. Quantum Hall Effect in a Spinning Disk Geometry
Syed Ali Raza
Supervisor: Dr. Pervez Hoodbhoy

2. Outline A brief Overview of Quantum Hall Effect Spinning Disk
Spinning Disk with magnetic Field
Percolation
Future investigations

3. Classical Hall Effect
F = v x B

4. Quantum Hall Effect 2-D system, perpendicular magnetic field
Quantized values of Hall Conductivity
σ = ne2/h
Quantised Landau Levels

5. Enormous Precision
Used as a standard of resistance
Does not depend on material or impurities

6. We first write our Hamiltonian
Define a Vector Potential
Solve it using many ways, e.g Operator approach

7. In terms of dimensionless variables
Cyclotron Frequency
Magnetic length

8. We define the Hamiltonian in terms of some operators

9. Degenerate States m degenerate states in each Landau level
the number of quantum states in a LL equals the number of flux quanta threading the sample surface A, and each LL is macroscopically degenerate.

10. Spinning Disk with no B field
Lagrangian
Hamiltonian
Lab Frame
Rotating Frame

11. Now we want to find the wavefunction for this Hamiltonian.
This has the form of the Bessel Equations
We take B = zero because otherwise there would be a singularity at r = zero.

12. where represents the nth root of the mth order Bessel function.
Bessel functions are just decaying sines and cosines.
We can also calculate the current for this spinning disk

13. Quantum Hall Effect in a disk Geometry
Hamiltonian
where
We make our equations dimensionless and get
Now we need to solve this to get the complete wavefunction.

14. We solve for U(r) using the series solution method and solve it exactly. After a lot of painful algebra, you get the following recursion relation:
And you can recover your energy relation from this recursion too

15. Getting the current
For a single electron

16. For more electrons

17. Spinning Disk but now with magnetic field
Lagrangian
Hamiltonian
Lab Frame
Rotating Frame

18. By Series Solution
Making them dimensionless and applying the wavefunction.
Applying the series solution method we get recursion relation
We can get the energies from this too

19. As you can see, because of the spinning there are no more m degenerate states in each landau level and now each m has an energy
The farther away from the centre, the more energetic they are
The series solution is very messy and tedious, so we try to do it with operators

20. By operator approach First we write our Hamiltonian
We set up our change of coordinates and operators

21. Substitute these in the Hamiltonian

22. Looks horrifying but gladly most of the things cancel out and we are left with
Plug in operators

24. We get our final Hamiltonian and energies

25. Percolation