Aharonov-Bohm Effect and Integer Quantum Hall Effect What is the relationship between Aharonov-Bohm effect and Integer Quantum Hall effect?
 A: 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:
$\sigma_H = \frac{j_{\phi}}{E} = \frac{e^2}{\hbar}$
