I have been recently reading X-G Wen's book on gauge theories. In chapter 6 he gave two examples of gauge theories showing confinement. The first one is a compact U(1) gauge theory in 2+1D, and the second one is a lattice U(1) theory in 3+1D. The treatments he gave for the two cases are different, for the 2+1D case he relied on duality a lot.

In both cases, in the confined phase, the gauge theories are also gapped. However, I do not immediately see the relation between the two concepts.

The main reason for my puzzle is, if a gauge theory is gapped, it should become short-ranged, and naively one should NOT expect the charges to strongly interact (in analogy with screening effect), let alone confinement. Hence the question in the title.

Update: I think I have figured out what is going on (see my comments below), but more comments/discussions are welcome!

  • $\begingroup$ On the other hand, I have also heard the following statement "all gapped gauge theories are topologically ordered". Any comments on this point are also welcome. $\endgroup$ – pathintegral Aug 11 '16 at 19:14
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    $\begingroup$ By definition, a gapped phase is one with a finite gap in the spectrum of the Hamiltonian between the vacuum state and the first excited state. A gapless phase, by contrast, has a continuous spectrum of states with arbitrarily small energies (e.g. corresponding to massless particles, when that interpretation makes sense). One can further classify various types of gapped and gapless phases with different qualitative features. A confining phase is one such example of a gapped phase. Another more familiar example is the Higgs phase. $\endgroup$ – Elliot Schneider Aug 13 '16 at 20:42
  • $\begingroup$ @user81003 Thanks. I do realize now the close relationship between confining phase and the Higgs phase. I guess my original question was, why a confined phase has to be gapped. The reason for my initial puzzle was, if a gauge theory is gapped, it should become short ranged, and naively one should NOT expect the charges to strongly interact. $\endgroup$ – pathintegral Aug 14 '16 at 2:58
  • $\begingroup$ My current understanding is that the gauge theory is gapped, but this is NOT saying that gauge charges are screened. Actually, it is the monopoles that are screened, and hence can proliferate. As a result, electric fields between gauge charges are required to be zero for the most part of space, and hence are "confined" within a thin tube. $\endgroup$ – pathintegral Aug 14 '16 at 3:01
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    $\begingroup$ gapped phase can be confined, and gapped phase can be deconfined as well. Gapped or not is based on the energy spectrum, but confined or not is based on the expectation value of some line operators. $\endgroup$ – user32229 Nov 13 '16 at 0:25

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