# A naive question on the Quantum Hall Effect(QHE) and the confinement in gauge theory?

The non-interacting 2D lattice QH system is described by the Hamiltonian $H=\sum t_{ij}e^{iA_{ij}}c_i^\dagger c_j+H.c$

My confusion is:

Does this imply that the $2D$ lattice QHE is described by the $2+1$ compact lattice $U(1)$ gauge theory? If it is, then according to the general conclusion " the $2+1$ compact lattice $U(1)$ gauge theory is always confinement ", thus, does this mean that the low-energy quasiparticles of $2D$ lattice QHE are not the free electrons?

I'm confused on this point, thank you very much.

• I'm a bit confused in so far as in the integer QH effect, the gauge field is a classical, non-dynamical degree of freedom, whereas in 2+1 lattice U(1) gauge theory, I would have guessed that the U(1) gauge field is dynamical and the two systems are thus fundamentally different (at least when you are referring to confinement/deconfinement transition I would presume that you are referring to a BKT like transition, which is definitely linked to the non-trivial topology U(1)/S1 of the dynamical degrees of freedom). – Jascha Ulrich Oct 10 '13 at 13:10

If you integrate out the fermions in the quantum Hall system (or the Chern insulator), you will end up with an effective $U(1)$ gauge theory, with a Chern-Simons term. The Chern-Simons term is originated from the non-zero Chern number of the occupied fermion bands, and reflects the Hall response of the system. This Chern-Simons term makes a great difference. With out it, the $U(1)$ compact lattice gauge theory is known to be in the confinement phase. But with the Chern-Simons term, a dynamic mass of photon is generated, so that the gauge fluctuation is gap away from the low-energy spectrum, and the theory is deconfined! Therefore the fermions (or quasiparticles) in the quantum Hall system are free.