Order-disorder
From Wikipedia, the free encyclopedia
 |
This article or section may require cleanup to meet Wikipedia's quality standards. Please help improve this article or replace this tag with a more specific message. This article has been tagged since December 2005. (help, talk) |
 |
The factual accuracy of this article or section is disputed. Please see the relevant discussion on the talk page. |
In quantum field theory and statistical mechanics, a system can be in two possible phases: an ordered phase and a disordered phase.
Let's give some examples here.
In the 2D Ising model, at low temperatures, there are two distinct pure states, one with the average spin pointing up and the other with the average spin pointing down. This is the ordered phase. At high temperatures, there is only one pure state with an average spin of zero. This is the disordered phase. The phase transition between the two is called the order-disorder phase transition.
To take another example, let's consider a real scalar field in 3+1 dimensions with a global O(3) symmetry and the field taking on values in a real 3D vector irreducible representation of O(3) (there's only one up to equivalence, it's not the pseudovector irrep). Then, if we only focus on finite energy superselection sectors, there are two regions in the parameter space. In the disordered region, the superselection sectors are labeled by irreducible reps of O(3) and the greatest upper bound of the energy spectrum in the superselection sectors labeled by a nontrivial rep is positive (i.e. there is a mass gap for charged particles). In fact, there is a mass gap even in the chargeless superselection sector. In the ordered region, the solitons are labeled by elements of the second homotopy group 
. Since there is a Z2 reflection (the internal O(3) "inversion"), the superselection sectors are labelled by a nonnegative integer (the absolute value of the topological charge) and furthermore, if this label is zero, there are infinitely many superselection sectors, each labelled by a O(2) irrep. Once again, there is a mass gap for the superselection sectors labeled by a nonzero integer (i.e. "topological solitons" are massive) but this time there is no mass gap for the all the superselection sectors labeled by zero (massless Goldstone bosons). Note the interesting fact that O(3) charges only make sense in the disordered phase and not at all in the ordered phase (because in a very handwaving way, we have a "charge condensate") and conversely, the topological charge only makes sense in the ordered phase and not at all in the disordered phase (in an even more handwaving way, we have a "topological condensate"). It's very telling that the very question of what charges are meaningful depend this much on the phase. If we approach the phase transition from the disordered side, the mass of the charges particles approaches zero and if we approach the phase transition from the ordered side, the mass of the topological solitions approaches zero as well.
To take another example, this time the standard model (let's forget all about QCD for the moment; it only complicates matters), the phase of parameter space we're in is the ordered phase and IF GUT theories are true (and this is a big if), then the superselection sectors are labeled by elements of Z (the magnetic monopole charge) and Z (the electric charge in multiples of 1/6), i.e. two integers. If there are no GUTs, then it's just one Z (the electric charge). But now, if we move to a different region in parameter space, the disordered phase, the superselection sectors are now labeled by SU(2)× U(1) irreps and IF GUTs are true, an element of Z for the magnetic monopole. Of course, this describes a fictitious world different from ours but we can get approximately the same thing in our universe by varying the temperature. Of course now, the superselection sectors are no longer labelled by charges because there are now infinitely many of them stretching to spatial infinity, but qualitatively, the same thing happens. Below the electroweak temperature, we're in the ordered phase where electric charges makes sense but not weak charges (after all, an electron keeps oscillating between a left handed electron with a nontrivial SU(2) charge and a right handed electron with a trivial SU(2) charge in the presence of a "weakly charged Higgs condensate"), but above the electroweak temperature, we're now in the disordered phase where SU(2)× U(1) charges makes sense.
Now let's look at the global flavour symmetry of QCD in the chiral limit where the masses of the quarks are zero (this is not really true in our universe where the up and down quarks have a tiny, but nonzero mass). Then, below the QCD temperature, we're in the ordered phase where SU(Nf) charges make sense where Nf is the number of flavors and above the QCD temperature, we're in the disordered phase where SU(Nf)×SU(Nf) and SU(3) charges makes sense.
