I am looking for some notes/reviews on confinement and Higgs phases suitable for Fermionic/Bosonic matter coupled to Abelian ($Z_2$ or $U(1)$ etc) gauge fields.

The purpose is to understand issues related to strongly interacting spin systems in 1,2,3 spatial dimensions, associated phase transitions, topological order etc. I am looking for help with references from which I can self-learn the things.

I will state couple of questions to convey the level of my lack of understanding.

  1. What is the relation between topological order and gauge theories?

  2. Why do people talk about topological order in a system described by Maxwell Lagrangian for U(1) gauge fields considering that for real QED around us, we do not talk about topological order?

  3. What is the difference between confinement phase and Higgs phase; how do I think of these things in a conceptually correct way (without violating Elitzur theorem)?

Some things I have found useful so far are these:

1) A question here in Phys.SE.

2) This freely available (through Springer) set of notes on confinement. Unfortunately, the stress here is on SU(3) gauge theories and QCD.


1 Answer 1


No doubt that a must read on this topic is the classic work by Fradkin and Shenker:


In particular, it was pointed out that for $Z_2$ gauge theories (and I believe for all $Z_n$) the confined phase and the Higgs phase are in fact smoothly connected. There is no sharp phase boundary between the two. More intuitively, the Higgs phase can be obtained by condensation of electric gauge charges, which means the $Z_2$ fluxes are confined, and Wilson loops satisfy area law. The confined phase is obtained by condensation of $Z_2$ fluxes (the magnetic charges) which confines the electric charges, and the electric flux loops satisfy area law. In the end of the day, it is just a trivial phase in both cases (without other symmetries). This is mostly evident in Kitaev's toric code model which realizes the $Z_2$ gauge theory, where the Higgs and the confined phases just correspond to all spins polarizing to different directions. One can of course continuously rotate the spins to any directions.

Transition between the deconfined and confined phases of $Z_2$ gauge theories has been later studied numerically. For example see http://journals.aps.org/prd/abstract/10.1103/PhysRevD.19.3682

Nowadays in the condensed matter theory community topological order is usually reserved for gapped systems. There is a good reason: for a gapped system, the long wavelength (much larger than the correlation length) or equivalently low-energy (much lower than the gap) physics is described by a topological field theory, of which discrete gauge theories are examples.

With this definition, U(1) Maxwell electrodynamics in 3+1d is not topologically ordered because it has no spectral gap. But this theory actually has many similarities with a topologically ordered state in 3+1d (i.e. $Z_2$ gauge theories). I can elaborate more later when I have time to write.


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