Syed Ali Raza
Supervisor: Dr. Pervez Hoodbhoy
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
2-D system, perpendicular magnetic
Quantized values of Hall Conductivity
σ = ne2/h
Enormous Precision, Used as a standard
of resistance
Does not depend on material or
impurities or geometry
Quantized Landau Levels
We first write our Hamiltonian
Define a Vector Potential
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.
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.
Making them dimensionless and applying the wavefunction.
Applying the series solution method we get recursion relation
We can get the energies from this too
First we write our Hamiltonian
We set up our change of coordinates and operators
Substitute these in the Hamiltonian
Looks horrifying but gladly most of the things cancel out and we
are left with
Plug in operators
We get our final Hamiltonian and energies
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
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?
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.
Bravais lattice vector
Translation operator
Translation operators do not commute
System is invariant under translation but the Hamiltonian is not as Vector Gauge changes
Number of magnetic flux passing through a unit cell
p and q are relatively prime
Magnetic bravais lattice; enlarged unit cell
Magnetic Translation operator
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
Bloch Wavefunction because of periodicity
Kubo’s Formula
And we get the TKNN invariant form of the Kubo’s Formula for quantised conductance
In a 2D periodic lattice
Take inner product to get wavefunction
Integral over the magnetic brillouin zone
We now define a vector potential like term
The integration is over the magnetic Brillouin zone
The magnetic Brillouin zone is a Torus T2 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.
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.
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
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.
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
Applying a magnetic field is a single perturbation, so we need to calculate the Green’s
function to the first order.
is proportional to
By Wicks theorem this becomes
is our perturbation
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
Our perturbation is of the form
So we get the current density
And the conductance
After Expansion
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.
Find Kubo for discrete case, on a lattice, like
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

Quantum Hall Effect in a Spinning Disk Geometry