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Simon Davenport

DPhil Student

I am a graduate student in theoretical condensed matter physics, working under the supervision of Prof. Steve Simon .

My research area is the fractional quantum Hall effect [FQHE]; for an introduction to the FQHE please go to the "Background on the FQHE" tab.

For more detailed information about my research and publications, go to the "Research" tab.

Background on the FQHE

The quantum Hall effect arises in the electronic structure of two-dimensional systems placed in high magnetic fields and is typically observed in high mobility GaAs quantum wells and heterojunctions. The impact of the magnetic field on the band structure is profound: degenerate bands called Landau levels (LLs) form. Such a system is parametrised by a quantity, known as the filling factor, which corresponds to the number of LLs filled with electrons or to the ratio of the number of electrons to the number magnetic flux quanta the sample. In the presence of weak disorder the Landau level structure becomes broadened: localised states occur when the chemical potential lies close to the Landau level centres and extended states occur in the Landau level tails.

At low temperatures (such that thermal excitations are smaller than the energy gap between LLs), the electronic properties of the sample can be dramatically altered by changing the applied magnetic field whilst keeping the chemical potential fixed, thus changing the position of the Landau levels relative to the chemical potential. Experimentally it is observed that as we sweep the value of the magnetic field then the component of the resistivity tensor rho_xx falls to zero over finite regions centered on a discrete set of filling factors, the principal series occurring for integer values of, corresponding to the chemical potential lying in the localised states. Simultaneously the value of rho_xy is fixed over the same region. This quantisation of the xy component of resistivity is exact to an extremely high degree of accuracy.

Intriguingly it is also observed that in extremely high magnetic fields the Landau levels also contain a sub structure, corresponding to the observation of these so called Hall plateaux at unexpected values of the filling factor, for example in the lowest Landau level 1/3, 2/5, 3/7 and so on (a modied pattern of odd denominator fractions occurs within each higher LL). The origin of the additional plateaux is the Fractional Quantum Hall effect (FQHE).

The FQHE occurs due to the presence of strong Coulomb interactions between electrons in the 2D electron gas in our semiconductor system: but it is not possible in general to exactly determine the wavefunctions describing a strongly interacting system of electrons! One option is to try to determine variational trial wavefunctions which attempt to minimise the Coulomb energy, whilst retaining the properties required to describe the observed electronic behavior of a quantum Hall system. One requires that our trial wavefunction fulfills the following criteria:

1. The trial wavefunction corresponds to a configuration of electrons in the 2D electron gas that minimises the overall electrostatic energy.

2. The trail wavefunction describes a system with a fixed filling factor and uniform electron density (in the ground state).

3. The trial wavefunciton describes an incompressible fluid of electrons.

References:

R. Prange and S.M. Girvin, The Quantum Hall Effect, (Springer-Verlag, NY (1987)).

[NOTE: this page is currently incomplete.]

Here's a list of research projects that I have worked on

Multi-Particle Pseudopotentials for Multi-Component Quantum Hall Systems

Published in Phys. Rev. B.

See also - ArXiv: cond-mat/1112.2709

Mathematica programs:

Program to calculate "primitive polynomials" for spatial wavefunctions.

Program to generate "generalised spin" wavefunctions and decompose these into spin eigenfunctions.

A draft version of the program documentation can be found here.

Abstract:

The Haldane pseudopotential construction has been an extremely powerful concept in quantum Hall physics --- it not only gives a minimal description of the space of Hamiltonians but also suggests special model Hamiltonians (those where certain pseudopotential are set to zero) that may have exactly solvable ground states with interesting properties. The purpose of this paper is to generalize the pseudopotential construction to situations where interactions are $ N $-body and where the particles may have internal degrees of freedom such as spin or valley index. Assuming a rotationally invariant Hamiltonian, the essence of the problem is to obtain a full basis of wavefunctions for $ N $ particles with fixed relative angular momentum $ L $. This basis decomposes into representations of $ SU(n) $ with $ n $ the number of internal degrees of freedom. We give special attention to the case where the internal degree of freedom has $ n=2 $ states, which encompasses the important cases of spin-1/2 particles and quantum Hall bilayers. We also discuss in some detail the cases of spin-1 particles ($ n=3 $) and graphene ($ n=4 $, including two spin and two valley degrees of freedom).

The Role of Spin in Some Fractional Quantum Hall Ground States

I gave this talk in Oxford on the 30/11/2011.

The completed paper can be found here.

An interesting recent experimental development: observation of a spin transition at filling 8/3.

We study fractional quantum Hall composite fermion wavefunctions at filling fractions $ \nu = 2/3 , 3/5, $ and $ 4/7 $. At each of these filling fractions, there are several possible wavefunctions with different spin polarizations, depending on how many spin-up or spin-down composite fermion Landau levels are occupied. We calculate the energy of the possible composite fermion wavefunctions and we predict transitions between ground states of different spin polarizations as the ratio of Zeeman energy to Coulomb energy is varied. Previously, several experiments have observed such transitions between states of differing spin polarization and we make direct comparison of our predictions to these experiments. For more detailed comparison between theory and experiment, we also include finite-thickness effects in our calculations. We find reasonable qualitative agreement between the experiments and composite fermion theory. Finally, we consider composite fermion state at filling factors $ \nu = 2+2/3 , 2+3/5 $, and $ 2+4/7 $. The latter two cases we predict to be spin polarized even at zero Zeeman energy.

Spinful Quantum Hall States at $ \nu=4/7 $

Work in progress. Here's the draft abstract:

We compare the characteristics of the $ k=2 $ non-abelian spin singlet (NASS) state with the series of composite fermion (CF) states occurring at filling $ \nu=4/7 $, specifically the states $ ^2 CF_{-4} $, $ ^2 CF_{(-3,-1)} $ and $ ^2CF_{(-2,-2)} $ (written in J. K. Jain's nomenclature). For the purposes of our comparison we calculate the ground state energy due to the simple Coulomb interaction in both the lowest Landau level (LLL) and the second Landau level. To be more realistic we also include the effects of a finite quantum well thickness. We have determined that for all of these interaction types, the NASS wavefunction lies very slightly lower in energy than any of the alternative CF states.

Adding Spin to the Gaffnian

Work in progress (in collaboration with Eddy Ardonne). Here's the draft abstract:

We will combine the properties of two quantum Hall wave functions. The first begin the nonunitary ‘Gaffnian’, the second the non-abelian spin-singlet (NASS) state.

View my research group