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

+ 2 like - 0 dislike

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.

This post imported from StackExchange Physics at 2014-03-09 08:41 (UCT), posted by SE-user K-boy
asked Oct 10, 2013 in Theoretical Physics by Kai Li (975 points) [ no revision ]
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).

This post imported from StackExchange Physics at 2014-03-09 08:41 (UCT), posted by SE-user Jascha Ulrich

1 Answer

+ 4 like - 0 dislike

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.

This post imported from StackExchange Physics at 2014-03-09 08:41 (UCT), posted by SE-user Everett You
answered Oct 10, 2013 by Everett You (660 points) [ no revision ]
@ Everett You Thanks a lot. So do you mean that as long as the gauge field is gapped, then the corresponding gauge theory is deconfined? And I read Wen's book of section 6.3.2 on page 265, as the title said,"The instanton effect in the (1 + 2)-dimensional U(1) gauge theory gives the gauge boson a finite energy gap and causes a confinement between the U(1) charges", so does this contradict to what you mean? I'm confused....

This post imported from StackExchange Physics at 2014-03-09 08:41 (UCT), posted by SE-user K-boy
@K-boy The confinement gap and the Chern-Simons gap are different gaps. A gap simply means the system is not critical. You can not tell from the gap whether the gauge theory is confined or not. Due to the Chern-Simons term, the instanton event is accompanied by charge pumping, which causes energy. So the instanton effect is suppressed, and hence the confinement is relieved.

This post imported from StackExchange Physics at 2014-03-09 08:41 (UCT), posted by SE-user Everett You

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