Topics for self-study¶
Simulations: Disorder, butterflies, and honeycombs¶
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!)
Pumping with disorder¶
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).
Now share your results:
For the third week we have these papers:
Topological States and Adiabatic Pumping in Quasicrystals (arXiv:1109.5983)¶
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 arXiv:1307.2577.
Spin Filtered Edge States and Quantum Hall Effect in Graphene (arXiv:cond-mat/0602645)¶
D. A. Abanin, P. A. Lee, L. S. Levitov
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 (arXiv:1201.4167)¶
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 (arXiv:0710.2806)¶
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 edge quasiparticles.
Bonus: Find your own paper to review!¶
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!