Significance of magnetic translation operator defined in fractional QHE's description What is the significance of the magnetic translation operator used in describing the Fractional Quantum hall effect? I was following Anthony Leggett's lecture video in which he defines these operators and describes their commutation relation in order to find the degeneracy of the ground state. Does it have any physical importance as such or is it just a mathematical construct in order to prove a result about the degeneracy of the ground state ?
 A: A quantum mechanical model in which the magnetic translation operators are observables is a charged particle moving on a two dimensional torus in the background of a uniform magnetic field perpendicular to the torus surface. Please, see for example the following  article  by E. Onofri.
The Hamiltonian is the magnetic Schrodinger operator and the ground state is the lowest Landau level. The full solution shows that the degeneracy of the lowest Landau level is equal to the magnetic flux through the torus surface area. Therefore, the magnetic flux must be quantized. This is the Dirac quantization condition. (There are many other ways to prove this result without the need of the full solution because Dirac quantization condition is a particular case of the index theorem).
A basis of wave functions of the lowest Landau level can be taken as the Jacobi theta functions, $\theta_{\nu}(z, \tau)$ where $z=x+iy$ is the complex coordinate on the torus, $\tau$ is proportional to the ratio between the torus generators and $\nu$ takes integer values between $1$ and the magnetic charge $N$.
The main difference of the Landau problem on the torus case from the plane is that the infinitesimal magnetic translation operators 
$\mathbf{p}-e\mathbf{A}$ are not observables because their action on  the wave functions lies outside the lowest Landau level. However, finite translations $e^{(\mathbf{p}-e\mathbf{A}).\mathbf{R}}$ are well defined if $e^{i|R|}$ is an $N$-th root of unity. 
This particular setting, however, cannot be easily implemented in lab at all, because it would require a net magnetic charge inside the torus and free magnetic charges have not been produced until now. 
However, this model can be translated into momentum space. Here the torus is a Brillouin zone of a $2D$ rectangular lattice. The restricted dynamics in one band can be described by an effective theory in which a Berry connection term is added to the Hamiltonian. Thus, this problem becomes analogous to the motion on the torus, but in momentum space. Here, in contrast to the real space, Berry's connections have nontrivial (fictitious) magnetic charges. Physical observables in the real space have analogous observables in momentum space. The Hall conductance is proportional to the magnetic charge of the Berry curvature, thus the Dirac quantization condition is responsible for the quantization of the Hall conductance.
