Quantum Hall Effect in a Spinning Disk Geometry - PowerPoint Presentation

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Quantum Hall Effect in a Spinning Disk Geometry

Syed Ali RazaSupervisor: Dr. Pervez Hoodbhoy

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Outline

A brief Overview of Quantum Hall EffectSpinning DiskSpinning 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

Slide3

Classical Hall Effect

F = v x B

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Quantum Hall Effect

2-D system, perpendicular magnetic fieldQuantized 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

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We first write our Hamiltonian

Define a Vector Potential

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Degenerate States

m degenerate states in each Landau levelthe 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

Solve for a spinning disk with out the magnetic fieldGet Bessel functions Solve for QHE in a disk geometry by both series solution and operator approach.

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Spinning Disk but now with magnetic field

LagrangianHamiltonian

Lab Frame

Rotating Frame

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By Series Solution

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

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By operator approach

First we write our HamiltonianWe 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

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

We don’t have to spin it ridiculously high frequenciesImpurities play an important role for the quantization of conductanceWe can mimic the broadening of peaks due to impurities and the broadening is in our controlEnergy splitting depends on the direction of rotationExplained by the orientation of spinning particles

Does spinning also affects Conductance?

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Kubo’s Formula for Conductance

Derive it using Perturbation theoryMagnetic field as the perturbationWhere 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.

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QHE on a magnetic Bravais Lattice

Bravais

lattice vector

Translation operator

Translation operators do not commute

System is invariant under translation but the Hamiltonian is not as Vector Gauge changes

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Number of magnetic flux passing through a unit cell

p and q are relatively prime

Magnetic

bravais

lattice; enlarged unit cell

Magnetic Translation operator

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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

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TKNN Invariance and Topology

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

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Take inner product to get

wavefunction

Integral over the magnetic

brillouin

zone

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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.

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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

e

verywhere. Anywhere where

wavefunction

u=0, there is an

ambiguity. You can multiply different things and still get the same result.

f

is not

n

ecessarily a continuous function.

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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

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As Torus is closed, the other stickman has to walk

a

long 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.

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Kubo’s Formula from Green’s theory

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

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Applying a magnetic field is a single perturbation, so we need to calculate the Green’s f

unction to the first order.

Where

i

s our perturbation

i

s proportional to

By Wicks theorem this becomes

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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

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Our perturbation is of the form

So we get the current density

And the conductance

After Expansion

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Kubo and Beyond, spinning disk with magnetic field

For the perturbation of spinning the disk, the conductance to second order is zeroWe 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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Future Plans

Find Kubo for discrete case, on a lattice, like MaitiSolve it for spinning disk with magnetic field and a 2D lattice structure by Green’s functionSimulate it and see the change in conductance due to spinning