Symmetry breaking domain walls

[JustinLevy]

Symmetry breaking "domain walls"

The only "spontaneously broken symmetry" that I can easily visualize, is cooling down a ferromagnetic material and having the spins randomly choose a direction to align. Since the choice is random, different regions will usually choose different directions, creating domains.

In theories where the vacuum spontaneously breaks a symmetry, are there also "domain walls" that form during non-equilibrium process of the symmetry breaking? What would these domains appear like, and can the theory be used to describe their evolution (or not really, since it wouldn't fit with our pertabative mathematical tools)?

 

[xepma]

Basically, any system which exhibits a spontaneously broken symmetry, which can be a gauge symmetry or some other type, has some way of allowing mismatches in its lowest energy, perfectly symmetric configurations. These mismatches carry some energy as they break the "optimal symmetry alignment".

So yes, there is a very richt study behind this. Such objects, which are formed due to a mismatch in the symmetry of the underlying degrees of freedom, are called topological defects. These defects can be localized (point-like or solitons) or stretch out in one or more directions (domain walls). They are called topological, because they can be classified using topological arguments of the corresponding symmetrygroup. This is also a way of understanding their stability: the defect arises due to a topological inequivalence between the state with a topological defect, and the state without one. On the other hand, it is possible for two defects to annihilate with each other, as they can "cancel each other out".

A famous example is the 't Hooft-Polyakov monopole, which predicts the existence of localized, monopole configurations in Yang-Mills theories. There are also the flux tubes you find in type II superconductors, which can be understood as vortices in the superconducting phase of the Cooper pairs. Another is the cosmic string -- yet to be discovered, and not to be confused with the strings in string theory --, which is the cosmic analog of the flux tube you find in superconductors. Such objects are believed to be formed during the cooldown after or during the inflational period of the early universe. Last but not least, there are also the vortices in the Bose-Einstein condensates, and the vortices in the quantum Hall effect.

 

[kikushiyo]

Hello JustinLevy,

As far as I concerned, I will just discuss phi^4 in broken phase. The answer is yes, there are domain walls (e.g. in the well-known d=4 for theory too).

When doing QFT for example using the effective potenial, the the non-perturbative aspects you expect are actually hidden in the flatness of the potential between the would-be minima of the tree-level potential (That's Maxwell construction).

An interesting point is that due to this non-perturbative effect, the perturbation theory fails to describe the inner region. It can be seen easily that the loop expansion is lost in this region due to the competiotn of two homogenous saddle points -signaling the presence of only one that is inhomogenous- and you obtain an imaginary part of the potential that may be interpreted as decay rate from the false vacuuim (phi=0) to the genuine one (phi=v).

Now turning to High Energy physics, you could wonder if the Higgs mechanism is actually perturbative. Indeed, you build the fluctuation to generate the mass of let's say fermions in the external region (phi>v) that should be perturbative (not seeing the competition in the internal region). But one the other hand the minimum lies at the exact location of the frontier between the two regions (the domain wall in question) so that you could feel non-perturbative fluctuations coming form the internal region as well.

So that this question that has been adressed decades ago has no clear answer as far as I know. We just hope that perturbation theory is valid on the external part of the broken potential... Let me mention that lattice proof of renormabizability of phi^4 in broken phase done by Luscher and Weisz had to ASSUME there was a perturbative relation between vertex function computed in 0 and the ones computed at the minimum...