A grand unification

In science, there is one thing that always makes me curious and awe. Roughly, I'd like to call it 'grand unification', which means a simple concept that connects various seemingly unrelated and independent phenomena which occur in distant disciplines. Such unification corroborates the conviction that, the universe has a common underlying mathematical structure. Here I talk of one more example of this.

This example is about fast symmetry breaking phase transitions (FSPT), in contrast with the usual cases where one talks of phase transitions near equilibrium states which may be described within hydrodynamic jargon. Up to my knowledge, no general effective theory for the moment has been established for FSPT. Despite this, it is possible to find out on some quantities very generic constraints which derive from the basic structure (such as symmetry and causality) of the first principles theory. Suppose a physical system is in equilibrium. Now one changes an experimental knob (e.g., pressure). This system shall then evolve away from its initial equilibrium state according to the corresponding dynamic theory, and shall eventually reach another new equilibrium state. However, what this final state might be should depend on the evolution and how the knobs are changed. An interesting thing is that, during this evolution some topological defects (which are textures that break the overall symmetry) shall form. Generally, it is expected that, as constrained by causality, which means no physical signal can travel faster than a typical velocity special to the system, two textures separated by a typical distance, say x, could not develop significant correlations during the evolution. Assume that a texture be characterized by 'direction vector'. Then, the causality indicates that, two distant textures should choose their directions independently in statistical sense. Therefore, all direction vectors may be found with certain textures because of the symmetry respected by the first principles model. Now, an interesting question is, how does x scale with the phase transition rate ?

In 1976, Kibble made an estimation [1], which is based on a cosmological model. It deals with cosmic strings, which are inherently stable topological objects that are expected to survive to today. Experimental verification of his idea is however quite difficult, since it is about the whole universe, which is so vast. A breakthrough came by Zurek [2], who tried Kibble's idea on Helium, which undergoes super-fluid transition at much low temperatures. And in such system, topological defects, which are known as fluxons and antifluxons by their winding numbers, could form as the transition is passed. Therefore, the helium system poses a remarkable realization of the big universe in this respect. To obtain the relation between x and transition rate, Zurek employed the dynamic Landau's theory, which was supposed to govern the temporary evolution. It turns out that, the relation follows a simple power law. Verification of this law came afterward. The latest one was done on superconductor [3]. For a good review see Ref.[4].

It is exciting to see that, the thing speculated about cosmos has its image on earth. I'd like to pose another question concerning FSPT, how does x scale with the dimension of the system ?

It may be worth pointing out that, as was emphasized by Anderson in his 'more is different' address, to deal with emergent phenomena, it is not enough to know just the first principles model (e.g., BCS model), which are usually not much useful in obtaining intuition and understanding. It is more useful to come up with an effective model (e.g., Landau's model), which concerns the quantities under imminent interest instead of those in the first principles model.

[1]J.Phys.A, 9:1387(1976)
[2]Nature, 317:505(1985)
[3]PRB, 80:180501(R), 2009
[4]Phys.Today, 60:47(2007)