Transport in the Quantum Hall Regime
N. d’Ambrumenil
The energy gap and an incompressible ground state
are essential components of all quantum Hall systems.
These states have attracted huge interest over many years
because they have turned up many unexpected properties including: fractionally charged excitations, fractional statistics (the excitations are between fermions
and bosons). A prediction is that the excitations may
even have nonAbelian statistics (interchanging excitations would return the system to a degenerate quantum
state).
µr
Tr
b
µr , Tr
2a
µl
Tl
µl , Tl
µl , Tl
Esp
−
−
µr , Tr
Esh
∆s
−a
a
Figure 1. Quantum Hall fluid in presence of a slowly varying background potential. The potential gives rise to regions
with quasiparticles (red) and quasiholes (blue) nucleated in
the quantum Hall fluid (white). Upper left is the symmetric case and upper right the case where the two quasihole
puddles are close to combining. Main picture: Band alignment across a saddle point in the potential. The (blue and
red) puddles serve as charge and heat baths which are weakly
coupled across the saddle. Temperature and potential differences, µl − µr and Tl − Tr , lead to net particle flows across
the system thereby allowing dissipation (the particles arriving in regions different µ and T requilibrate in the new charge
reservoirs/heat baths.)
Estimates of this gap have usually been found by fitting the temperature-dependence of the longitudinal resistance of the highest mobility samples to the standard
Arrhenius form [1]. However understanding of how the
measured activation gap relates to the intrinsic gap expected from theory has been held up by the lack of a detailed microscopic model of the effect of disorder on the
dissipative transport. We have recently introduced [2] a
theoretical model of dissipation which explains many of
the observations (see figure).
The thermopower of a homogeneous electron fluid is
related to the entropy per particle [3, 4]. In quantum
Hall systems, this information may be hard to extract
given that the charge and heat currents are known to be
inhomogeneous with a large role played by edge effects
which may be different for heat and charge transport.
However, measurements in the ‘edgeless’ Corbino geometry are expected to get round many of the complications
associated with inhomogeneous current distributions [5].
The aim of this project is to develop a model to account
for the temperature dependence of the various transport
coefficients with the aim of establishing which properties
of the quantum Hall fluid are directly measurable in experiment. The simple model [2] needs to be extended to
study thermally induced charge flows and the the flow of
heat. Direct measurements of the the entropy which are
in principle possible in homogeneous systems would be
the most exciting possibility as the nonAbelian excitations imply large degeneracies and hence entropy at very
low temperatures.
One further avenue to explore is whether the energy
to create a quasiparticle-quasihole excitation is different
in the neighbourhood of a saddle point—as experimental
data seem to suggest. This would involve solving numerically for the ground and lowest excited states of the
Hamiltonian of a system near a boundary.
[1] C. R. Dean, A study of the fractional quantum hall energy
gap at half filling, Ph.D. thesis, McGill (2009).
[2] N. d’Ambrumenil, B. Halperin, and R. Morf, Phys. Rev.
Lett. 106, 126804 (2011).
[3] N. Cooper, B. Halperin, and I. Ruzin, Phys. Rev. B 55,
2344 (1997).
[4] K. Yang and B. Halperin, Phys. Rev. B 79, 115317 (2009).
[5] Y. Barlas and K. Yang, “Thermopower of quantum hall
states in corbino geometry as a measure of quasiparticle
entropy,” Cond-mat:1202.4102.