What is the relationship between Aharonov-Bohm effect and Integer Quantum Hall effect?
[Physics] Aharonov-Bohm Effect and Integer Quantum Hall Effect
berry-pancharatnam-phasequantum mechanicsquantum-hall-effect
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In a naive picture, the zero resistance in superconductivity arises from the macroscopic superposition of the wavefunction of the $N$ Cooper pairs, which are bosons. Below the critical temperature $T_c$, the ensemble of bosons condensate into a Bose-Einstein condensate and the response to an electric field is dissipationless because of macroscopic coherence.
In the case of the integer quantum Hall effect, the zero resistance is due to the momentum-locking of the eigenstates in the direction parallel to the electric field. This can be understood as a single particle phenomenon which appears thanks to the external applied magnetic field $\mathbf{B}$: states in each edge of the $2$D sample move in opposite directions and backscattering between edges is forbidden except at the percolation transition inbetween the Hall plateaux.
As it is said below in the comment by wsc, there is also an important distinction between both systems from the experimental point of view. In a superconductor below $T_c$, the resistance is strictly zero while in the Hall plateaux there is always a residual resistance at $T>0$. In the latter, one cannot overemphasize the crucial role of disorder for the drop of longitudinal resistance and quantization of the Hall component.
Moreover, the dimensionality plays an important role in the tensorial nature of the resistance tensor (in $2$D, one can loosely identify resistance and resistivity due to dimensionality reasons). In this case, inverting the relation $\mathbf{E}=\bar{\rho}\mathbf{j}$ shows that $\rho_{xx} \propto \sigma_{xx}$ and a small resistivity in the longitudinal direction also involves a small non vanishing conductivity.
I'm pretty sure the answer to that is an emphatic yes: The Aharonov-Bohm effect has been proven many, many times experimentally. My recollection is that it is described as experimentally proven in the Feynman Lectures from way back in the late 1960s. Looking...
Ah, here it is: Volume II, Chapter 15, Section 15-5, "The vector potential and quantum mechanics" (starting on page 15-8), page 15-12, second paragraph from the bottom. The name is misspelled as "Aharanov" in some older editions, incidentally.
That reference likely not a proof, but I know I've seen other very specific, detailed, and far more recent papers and discussion of the effect, including some very nice ones using small lithographically etched circles. I liked that one especially because it make the idea of Aharonov-Bohm as a type of displacement vector easier to visualize. Looking again...
I don't believe this is the same paper I was thinking of, but it's 2008 and pretty similar in testing strategy. Definitely worth a look if you are interested, and likely has relevant references also:
Mesoscopic decoherence in Aharonov-Bohm rings, by A.E. Hansen, A. Kristensen, S. Pedersen, C.B. Sørensen, and P.E. Lindelof. The Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark (February 1, 2008).
And two more for completeness! First, this most excellent paper found by @JoeHobbit (see his comment below) looks like it may be "the paper" that in 1986 -- much later than the Feynman mention -- first provided convincing proof of the existence of Aharonov-Bohm:
Experimental confirmation of Aharonov-Bohm effect using a toroidal magnetic field confined by a superconductor, by Nobuyuki Osakabe, Tsuyoshi Matsuda, Takeshi Kawasaki, Junji Endo, and Akira Tonomura. Phys. Rev. A 34, 815–822 (1986). This classic paper is also available here through scribd, and also here.
For a much more recent example of just how mature this area is these days, there is this 2013 (!) paper on using graphene (I love graphene) to explore Aharonov-Bohm oscillations:
Transport properties of two finite armchair graphene nanoribbons, by Luis Rosales and Jhon W. González. Nanoscale Research Letters 2013, 8:1 doi:10.1186/1556-276X-8-1 (2 January 2013).
(Kudos to Springer for making that and similar papers truly open, by the way!)
2013-02-18. Alas, I've not received any response from Dr Aharonov, though to be honest I thought it unlikely. Worth a try, though; I'd have loved to hear his view.
After going over Timothy Boyer's paper, and in particular after seeing that in 2006 he proposed some specific experiments (inaccessible without paying) to distinguish between relativistic effects and a true Aharonov-Bohm effect, my position is pretty simple: If his experimental setup looks reasonable, someone should should try Boyer's experiment and see what they get.
There is nothing unreasonable about the sequence of first noting possible modeling errors (which is all Boyer is really doing, and doing it pretty well from what I can see), next giving an alternative modeling option that addresses the concerns of the author, and then suggesting an actual experiment to distinguish between the predictions.
Why not just do the experiment? Seems like a good way either to firm up the nature of all Aharonov-Bohm results, or add some interesting new discussion if the experiment produces anything unexpected.
If Boyer's experiment is flawed in concept (again, I cannot see that paper!), can anyone on this group who has seen it say why?
Best Answer
There is a famous argument by R. Laughlin Phys. Rev. B 23, 5632–5633 (1981) explaining the integer quantum Hall effect based on the Aharonov-Bohm effect. This argument is explained in the following lecture notes by Manfred Sigrist (page 70, please see figure 3.17). The argument goes as follows:
Consider a system of electrons moving on a two dimensional annulus subject to a very large magnetic field so, they are constrained to the (degenerate) lowest Landau levels, because the energy they need to shift to an excited level is very large. In addition suppose that an electric field $E$ is applied in the radial direction Suppose that an increment of a uniform magnetic flux is applied in the annulus hole. According to Aharonov-Bohm, if this increment is an integer multiple of $\frac{\hbar c}{e}$, the physics should remain the same because in this case the increment can be removed by a gauge transformation.
Suppose, now that this increment is applied adiabatically, in this case the (mean) radius of the Landau level will increase adiabatically and when the flux becimes an integer, the electron will necessarily occupy the next lowest Landau level of the original Lagrangian because the Lagrangian is the same up to a gauge transformation. In particular, the net change in the magnetic and electric energies should be zero. The net change in the magnetic energy is:
$ \delta E_{M} = \frac{e}{mc} \mathbf{p}.\mathbf{\delta A} = \frac{1}{c} I_{\phi} \delta \Phi = \frac{1}{c} j_{\phi} b \frac{\hbar c}{e} $
Where $I_{\phi}$ is the angular current and $\delta \Phi$ is the flux increment $j_{\phi}$ is the angular current density and $b$ is the average radial distance between two Landau Levels. The electric energy is: $ \delta E_{E} =-e E b$ Equating the two contributions one gets the contribution of a single electron to the hall conductivity:
$\sigma_H = \frac{j_{\phi}}{E} = \frac{e^2}{\hbar}$