```Syed Ali Raza
Supervisor: Dr. Pervez Hoodbhoy
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A brief Overview of Quantum Hall Effect
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Spinning Disk
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Spinning Disk with magnetic Field
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Kubo’s Formula for Conductance
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QHE on a magnetic Bravais Lattice
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TKNN Invariance and Topology
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Kubo’s Formula from Green’s theory
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Kubo and Beyond, spinning disk with magnetic field
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Future Plans
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F=vxB
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2-D system, perpendicular magnetic
field
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Quantized values of Hall Conductivity
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σ = ne2/h
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Enormous Precision, Used as a standard
of resistance
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Does not depend on material or
impurities or geometry
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Quantized Landau Levels
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We first write our Hamiltonian
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Define a Vector Potential
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m degenerate states in each Landau level
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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.
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We change to polar coordinates
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Solve for a spinning disk with out the
magnetic field
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Get Bessel functions
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Solve for QHE in a disk geometry by both
series solution and operator approach.
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Lagrangian
Hamiltonian
Rotating
Frame
Lab
Frame
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Making them dimensionless and applying the wavefunction.
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Applying the series solution method we get recursion relation
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We can get the energies from this too
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First we write our Hamiltonian
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We set up our change of coordinates and operators
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Substitute these in the Hamiltonian
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Looks horrifying but gladly most of the things cancel out and we
are left with
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Plug in operators
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We get our final Hamiltonian and energies
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Degeneracy is lifted and there is broadening of peaks of Landau
level Energies
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We don’t have to spin it ridiculously high frequencies
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Impurities play an important role for the quantization of
conductance
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We can mimic the broadening of peaks due to impurities and the
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Energy splitting depends on the direction of rotation
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Explained by the orientation of spinning particles
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Does spinning also affects Conductance?
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Derive it using Perturbation theory
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Magnetic field as the perturbation
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Where α and β represent the states below and above
the Fermi level respectively.
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With perturbation of spinning, it’s zero to first order
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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
Using
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.
Where
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.
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Find Kubo for discrete case, on a lattice, like
Maiti
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Solve it for spinning disk with magnetic field
and a 2D lattice structure by Green’s function
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Simulate it and see the change in
conductance due to spinning
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