1. Syed Ali Raza Supervisor: Dr. Pervez Hoodbhoy

2. A brief Overview of Quantum Hall Effect Spinning Disk Spinning Disk with magnetic Field Kubo’s Formula for Conductance QHE on a magnetic Bravais Lattice TKNN Invariance and Topology Kubo’s Formula from Green’s theory Kubo and Beyond, spinning disk with magnetic field Future Plans

3.  F = v x B

4.  2-D system, perpendicular magnetic field  Quantized values of Hall Conductivity  σ = ne 2 /h  Enormous Precision, Used as a standard of resistance  Does not depend on material or impurities or geometry  Quantized Landau Levels

5.  We first write our Hamiltonian  Define a Vector Potential

6.  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.

7.  We change to polar coordinates  Solve for a spinning disk with out the magnetic field  Get Bessel functions  Solve for QHE in a disk geometry by both series solution and operator approach.

8.  Lagrangian  Hamiltonian Lab Frame Rotating Frame

9.  Making them dimensionless and applying the wavefunction.  Applying the series solution method we get recursion relation  We can get the energies from this too

10.  First we write our Hamiltonian  We set up our change of coordinates and operators

11.  Substitute these in the Hamiltonian

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

13.  We get our final Hamiltonian and energies

14.  Degeneracy is lifted and there is broadening of peaks of Landau level Energies  We don’t have to spin it ridiculously high frequencies  Impurities play an important role for the quantization of conductance  We can mimic the broadening of peaks due to impurities and the broadening is in our control  Energy splitting depends on the direction of rotation  Explained by the orientation of spinning particles  Does spinning also affects Conductance?

15.  Derive it using Perturbation theory  Magnetic field as the perturbation  Where α and β represent the states below and above the Fermi level respectively.  With perturbation of spinning, it’s zero to first order  Gets two complicated for both perturbations.

16. Bravais lattice vector Translation operator Translation operators do not commute System is invariant under translation but the Hamiltonian is not as Vector Gauge changes

17. Number of magnetic flux passing through a unit cell p and q are relatively prime Magnetic bravais lattice; enlarged unit cell Magnetic Translation operator

18. There is a phase change when you go around the magnetic cell’s boundary p is the number of flux quanta passing through a magnetic unit cell p is a topological invariant

19. Bloch Wavefunction because of periodicity Kubo’s Formula Using And we get the TKNN invariant form of the Kubo’s Formula for quantised conductance In a 2D periodic lattice

20. Take inner product to get wavefunction Integral over the magnetic brillouin zone

21. We now define a vector potential like term The integration is over the magnetic Brillouin zone The magnetic Brillouin zone is a Torus T 2 rather than a rectangle in k space As the Torus has no boundary, applying stoke’s theorem will give zero for the integral above, if A is well defined all over the Torus.

22. But A is not defined well over the Torus and we would try to understand it Both of them satisfy Schrodinger equation All physical quantities remain the same under this transformation Non Trivial Topology arises when the phase of the wavefunction cannot be uniquely determined in the entire magnetic Brillouin zone But f is not well defined everywhere. Anywhere where wavefunction u=0, there is an ambiguity. You can multiply different things and still get the same result. f is not necessarily a continuous function.

23. Suppose u vanishes here, so we isolate the patch There is a phase mismatch at the boundary Apply Stoke’s theorem to both of them separately

24. As Torus is closed, the other stickman has to walk along the boundary In the opposite direction. n is an integer as we showed before in the slides; that the integral of the phase over the magnetic Brillouin zone gives an integer. Also known as the Chern number. Conductance is quantized and Topologically protected.

25. Green’s function, important as it only considers the linked diagrams, perturbation Theory by Feynman Diagrams We first carefully calculate the first order m=0 of the Green’s function, important basic unit

26. Applying a magnetic field is a single perturbation, so we need to calculate the Green’s function to the first order. Whereis our perturbation is proportional to By Wicks theorem this becomes

27. Now we have the first order in terms of the zeroth order which we have already calculated We have to solve the following structure For any body operator

28. Our perturbation is of the form So we get the current density And the conductance After Expansion

29. For the perturbation of spinning the disk, the conductance to second order is zero We now do it for two perturbations, have to solve second green’s function Now order will also matter, of which perturbation (spinning or B field) came first, so sum over both diagrams. Again, when we expand, the first term goes to zero, but we get a nice second term.

31.  Find Kubo for discrete case, on a lattice, like Maiti  Solve it for spinning disk with magnetic field and a 2D lattice structure by Green’s function  Simulate it and see the change in conductance due to spinning