Simulations: Disorder, butterflies, and honeycombs¶

As usual, start by grabbing the notebooks of this week (w3_pump_QHE). They are once again over here.

There are really plenty of things that one can study with the quantum Hall effect and pumps. Remember, that you don't need to do everything at once (but of course all of the simulations are quite fun!)

Grab the simulations of the Thouless pump, and see what happens to the pump when you add disorder. Try both the winding in a pump with reservoirs attached, and the spectrum of a closed pump. Can you explain what you observe?

Take a look at how we calculate numerically the spectrum of Landau levels in the Laughlin argument chapter.
We were always careful to only take weak fields so that the flux per unit cell of the tight binding lattice is small.
This is done to avoid certain notorious insects, but nothing should prevent you from cranking up the magnetic field and seeing this beautiful phenomenon.

Plot the spectrum of a quantum Hall layer rolled into a cylinder at a fixed momentum as a function of $B$ as $B$ goes to one flux quantum per unit cell, so in lattice units $B = 2\pi$. Bonus (requires more work): attach a lead to the cylinder, calculate pumping, and color the butterfly according to the pumped charge.

Take a look at how to implement a honeycomb lattice in Kwant tutorials, and modify the Hall bar from the Laughlin argument notebook to be made of graphene. Observe the famous unconventional quantum Hall effect.

Bonus: See what happens to the edge states as you introduce a constriction in the middle of the Hall bar. This is an extremely useful experimental tool used in making quantum Hall interferometers (also check out the density of states using the code from the edge states notebook).

MoocSelfAssessment description

In the live version of the course, you would need to share your solution and grade yourself.

Now share your results:

Discussion Quantum Hall effect is available in the EdX version of the course.

Yaacov E. Kraus, Yoav Lahini, Zohar Ringel, Mor Verbin, Oded Zilberberg

The unrelated discoveries of quasicrystals and topological insulators have in
turn challenged prevailing paradigms in condensed-matter physics. We find a
surprising connection between quasicrystals and topological phases of matter:
(i) quasicrystals exhibit nontrivial topological properties and (ii) these
properties are attributed to dimensions higher than that of the quasicrystal.
Specifically, we show, both theoretically and experimentally, that
one-dimensional quasicrystals are assigned two-dimensional Chern numbers and,
respectively, exhibit topologically protected boundary states equivalent to the
edge states of a two-dimensional quantum Hall system.We harness the topological
nature of these states to adiabatically pump light across the quasicrystal. We
generalize our results to higher-dimensional systems and other topological
indices. Hence, quasicrystals offer a new platform for the study of topological
phases while their topology may better explain their surface properties.

Hint: Topological pumping can be used to characterize quasicrystals too! Whether this is really unique to quasicrystals is debated though http://arxiv.org/abs/1307.2577v2

Spin Filtered Edge States and Quantum Hall Effect in Graphene

Electron edge states in graphene in the Quantum Hall effect regime can carry
both charge and spin. We show that spin splitting of the zeroth Landau level
gives rise to counterpropagating modes with opposite spin polarization. These
chiral spin modes lead to a rich variety of spin current states, depending on
the spin flip rate. A method to control the latter locally is proposed. We
estimate Zeeman spin splitting enhanced by exchange, and obtain a spin gap of a
few hundred Kelvin.

Hint: Quantum Hall effect applies beyond parabolic dispersions with interesting twists. Figure out what different features arise from other cases.

Spin and valley quantum Hall ferromagnetism in graphene

Andrea F. Young, Cory R. Dean, Lei Wang, Hechen Ren, Paul Cadden-Zimansky, Kenji Watanabe, Takashi Taniguchi, James Hone, Kenneth L. Shepard, Philip Kim

In a graphene Landau level (LL), strong Coulomb interactions and the fourfold
spin/valley degeneracy lead to an approximate SU(4) isospin symmetry. At
partial filling, exchange interactions can spontaneously break this symmetry,
manifesting as additional integer quantum Hall plateaus outside the normal
sequence. Here we report the observation of a large number of these quantum
Hall isospin ferromagnetic (QHIFM) states, which we classify according to their
real spin structure using temperature-dependent tilted field magnetotransport.
The large measured activation gaps confirm the Coulomb origin of the broken
symmetry states, but the order is strongly dependent on LL index. In the high
energy LLs, the Zeeman effect is the dominant aligning field, leading to real
spin ferromagnets with Skyrmionic excitations at half filling, whereas in the
`relativistic' zero energy LL, lattice scale anisotropies drive the system to a
spin unpolarized state, likely a charge- or spin-density wave.

Hint: An experiment detecting the interesting consequences of coexistence of quantum Hall and ferromagnetism in graphene.

Direct measurement of the coherence length of edge states in the Integer
Quantum Hall Regime

P. Roulleau, F. Portier, P. Roche, A. Cavanna, G. Faini, U. Gennser, D. Mailly

We have determined the finite temperature coherence length of edge states in
the Integer Quantum Hall Effect (IQHE) regime. This was realized by measuring
the visibility of electronic Mach-Zehnder interferometers of different sizes,
at filling factor 2. The visibility shows an exponential decay with the
temperature. The characteristic temperature scale is found inversely
proportional to the length of the interferometer arm, allowing to define a
coherence length $\l_\phi$. The variations of $\l_\phi$ with magnetic field are
the same for all samples, with a maximum located at the upper end of the
quantum hall plateau. Our results provide the first accurate determination of
$\l_\phi$ in the quantum Hall regime.

Hint: Aharonov-Bohm interference using quantum hall edgequasiparticles.

Do you know of another paper that fits into the topics of this week, and you think is good?
Then you can get bonus points by reviewing that paper instead!

MoocSelfAssessment description

In the live version of the course, you would need to share your solution and grade yourself.

Do you have questions about what you read? Would you like to suggest other papers? Tell us:

Discussion Quantum Hall effect is available in the EdX version of the course.