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Phase transitions are often detected by an order parameter - some quantity which is zero in the "disordered" phase, but becomes non-zero when order is established. For example, if the phase transition is accompanied by a breaking of a global symmetry, an order parameter can be any quantity which transforms non-trivially so that it averages to zero in the disordered phase.

Phases not characterized by their global symmetry are more tricky. For confinement there are well-known order parameters: the area law for the Wilson loop, the Polyakov loop (related to breaking of the center symmetry), and the scaling of the entropy with N in the large N limit.

My question is then about the Higgs phase, which is usually referred to (misleadingly in my mind) as spontaneous breaking of gauge "symmetry". More physical (among other things, gauge invariant) characterization of this phase would be in terms of some order parameter. What are some of the order parameters used in that context?

(One guess would be the magnetic duals to the quantities characterizing confinement, but there may be more).

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One thought: certainly for a Higgs in the fundamental, I would think an area law for the 't Hooft loop would be a good criterion. But for an adjoint Higgs, it shouldn't (unless I'm more confused than I realize). I think this is why it's somewhat standard to distinguish a "Higgs branch" from a "Coulomb branch", right? –  Matt Reece Feb 25 '11 at 23:05
Yeah, by Higgs branch I mean the case where the gauge invariance is completely broken, but I'd be also interested to see what happens if you leave some light gauge bosons around. I'm mainly wondering if there is something I am missing... –  user566 Feb 25 '11 at 23:08
Silly thought, but wouldn't the order parameters just be the particle masses generated by the Higgs mechanism? –  user346 Feb 26 '11 at 2:48
Sort of, one still has to be careful to phrase things in a gauge invariant way. So for example you can talk about gap in the spectrum, or the effective range of the force between probes, or something else that encodes this fact in a physical quantity. Which is sort of my question... –  user566 Feb 26 '11 at 3:25
There are certainly some theories in which confinement and Higgs are not separated by a sharp phase transition, but there are theories where they are. In those, there ought to be order parameter(s) which distinguishes them. –  user566 Feb 26 '11 at 4:21
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This is a real, and often encountered problem in general in condensed matter. As the comments in Lubos' answer discusses, it's not easy to invent an "order parameter" which unambiguously yields the correct phase since symmetries can't really be broken, and 2nd order correlators tend to display quantum/thermal fluctuations anyway. In condensed matter theory, a frequently used construct is the Off-Diagonal Long Ranged Order, $\lim_{r \rightarrow \infty} \langle \psi(0)\psi(r) \rangle$, which gives a measure of correlation. In practice however, real systems are finite and it is hard to take the limit physically. Anthony Leggett has though long and hard about this problem in the context of cold atoms and quantum condensation, and suggests something based around the single particle density matrix; it is unclear whether this generalises into things like QCD.

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Dear Moshe, your question seems to demand the answer to be very complicated and unexpected and the obvious answer seems to be forbidden in between the lines except that I think that the obvious answer is right.

The Higgs field's vacuum condensate is the order parameter of the vacuum in the electroweak theory. And if you define it in a gauge-invariant way, $$\sqrt{H^\dagger H},$$ then it is gauge-invariant, too. Obviously, this quantity is still the same $v$ we all know and love. This fact, that the Higgs field is the order parameter, lies behind the fact that the Higgs mechanism uses the same maths as Landau's theory of phase transitions.

The vev can't be measured "directly" as a mass of something but it may still be measured indirectly - e.g. from the masses it gives to other particles (because the couplings may be measured, too).

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I know how the Higgs mechanism works in the electroweak theory. But, suppose you can indirectly calculate (or measure) any physical quantity you want in a theory you are less familiar with, say by some dual formulation or something. What will convince you the theory is in the Higgs phase? –  user566 Feb 26 '11 at 6:28
Otherwise, if you want to distinguish the "Higgs phase" qualitatively from "other phases", without specifying a symmetry, then I think it cannot be done. There is nothing qualitatively different about "Higgs phases". Take heterotic strings on a circle. The generic gauge group is $U(1)^{16}$. It can be called a Higgsed $SO(32)$ so the point with $SO(32)$ is a non-Higgsed phase. However, the point with $E_8\times E_8$ has an equally big unbroken symmetry, so it is also "unHiggsed" although it would clearly be "spont. broken" if you defined the latter relatively to $SO(32)$. –  Luboš Motl Feb 26 '11 at 6:40
After all, the Higgs phase is often continuously connected with the confinement phase, see Banks Rabinovici 1979 and physics.stackexchange.com/questions/5785/… –  Luboš Motl Feb 26 '11 at 6:41
I think you can imagine theories where this kind of information doesn't discriminate phases. E.g., in Yang-Mills with an adjoint scalar, a vev for the gauge-invariant $Tr(\phi^2)$ might mean $\phi$ got a vev and the theory is in a Coulomb phase, or it might mean the theory was confining and, just as other gauge-invariant operators like $Tr(F^{\mu\nu}F_{\mu\nu})$ do, the composite operator got a vev of order the confinement scale. Right? But, it might be that in all the cases where "Higgs phase" makes sense, the confinement/Higgsing distinction breaks down, so such a vev answers the question. –  Matt Reece Feb 26 '11 at 6:44
I'm saying that knowing the expectation values of gauge-invariant operators is not necessarily, by itself, diagnostic of what phase a theory is in. If all that you tell me is I'm looking at a gauge theory and $Tr(\phi^2)$ is nonzero for adjoint $\phi$, I don't know if you've given me a theory that is in a Coulomb phase or a confining phase without asking additional questions (like whether there are massless vectors around). On the other hand, if you tell me that I have a theory with $Tr(H^\dagger H)\neq 0$ with $H$ in the fundamental, I can't think of any interpretation other than Higgsing... –  Matt Reece Feb 26 '11 at 6:58
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