# Topological Phases and Confinement

I recently attended a talk in which the speaker defined a topological phase as "A phase which has a gap above the ground state for bulk excitations in the thermodynamic limit." I am interested in what sense then can we think of confinement in non-Abelian Yang-Mills theories as topological phases.

What I'm looking for are the analogies; what would the thermodynamic limit and the bulk excitations mean if we were talking about a YM theory (QCD, for instance). The thermodynamic limit is the number of particles $N\to\infty$, which I suppose we can think of as the number of Feynman diagrams (order of the loops) growing as large as possible. "The bulk" seems a bit more vague (which is perhaps because of my definition), but it seems like bound states of quarks is the appropriate notion for that.

So, is it possible to (correctly) say something like "QCD is a topological phase for the standard model"? If not, is there a clear reason why this is not the case?

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You can see Aharony et al's paper on Hagedorn deconfinement transition. (in general see Kyriakos's PhD papers) They consider the large N_c limit of free Yang-Mill's theory on compact space-time. That theory seems to be permanently confined. –  user6818 May 28 '13 at 19:34

Indeed, a topological phase is "a phase which has a gap above the ground state for bulk excitations in the thermodynamic limit." But some topological phases defined by such a definition may have trivial topological order, while other topological phases defined by such a definition may have non-trivial topological order. (For a definition of topological order see http://en.wikipedia.org/wiki/Topological_order ).

The confined phase of non-Abelian Yang-Mills theories is not a well defined term. Different cut-offs can give raise to different confined phases of non-Abelian Yang-Mills theories. Some of those confined phases have non-trivial topological order while other confined phases have trivial topological order.

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Have you seen this recent paper arxiv.org/abs/1301.7072 ? I briefly looked at it when it came out and it seemed very interesting. I guess its very relevant to this question. –  Heidar May 29 '13 at 2:12
There is a wrong statement in the abstract of the paper arxiv.org/abs/1301.7072. Topological insulators/superconductors in fact are not topologically ordered phases. –  Xiao-Gang Wen May 29 '13 at 5:33
How do you know what the topological order of the confined phase of non-Abelian YM theory? –  levitopher May 30 '13 at 13:59
One way is to measure ground state degeneracy on torus and their non-Abelian geometric phases. See en.wikipedia.org/wiki/Topological_order –  Xiao-Gang Wen May 31 '13 at 5:12