# Signatures of universality beyond the mean number of topological defects

No matter what you build or how much care you take building it, it would have defects. It seems it is a rule of nature: every time chaos gives way to order, from matter itself coming into existance in the very early universe to the formation of a crystal in a modern laboratory, defects appear. But defects are not always a bad thing, as semiconductor devices prove. In any case, when a phase transition occurs, understanding how defects appear is most important both in basic science and its technological applications.

Thomas Kibble was not only famous for his involvement in the theoretical development of the Higgs mechanism. His research on phase transitions and topological defects is most significant. Actually, the Kibble-Zurek mechanism (KZM) is a paradigmatic theory to describe the dynamics across both classical continuous phase transitions.

KZM describes the non-equilibrium dynamics and the formation of topological defects in a scenario of spontaneous symmetry breaking: a system which is driven through a continuous phase transition at finite rate. It is named after Kibble, who pioneered the study of domain structure formation in the early universe, and Wojciech H. Zurek, who related the number of defects it creates to the critical exponents of the transition and to how quickly the critical point is traversed.

According to KZM, defects appear when a system in a disordered phase is thrown out of equilibrium into an ordered phase. Order does not appear in all the system at once, but in certain points, that then grow to form domains of volume that keep growing, eventually coming together. KZM predicts that the density of topological defects obeys a universal power-law with respect to the cooling rate (quench time) of the system. This power law behavior with the quench time, initially derived for classical systems, similarly describes the dynamics across quantum phase transitions.

KZM has been extended to a variety of scenarios including nonlinear quenches, long-range interactions, and inhomogeneous phase transitions in both classical and quantum systems. The KZM model has also been experimentally tested in a in a wide variety of platforms. Still, despite this progress, the counting statistics of defects has not gone beyond its mean number.

Now, a team of researchers has focused ^{1} on signatures of universality beyond the mean number of topological defects and showed that the full counting statistics of topological defects is actually universal.

The same way the formation of vortices has been demonstrated by merging independent Bose-Einstein condensates, topological defects may form at the interface between multiple domains. Here lies the novelty of the approach: in framing the Kibble-Zurek picture as one in which a topological defect may nucleate at the junction of ordered domains. Or it may not. ^{2}. Besides, the formation of topological defects at different locations is assumed to be independent and in each case the event of formation can be associated with a Bernoulli random variable.

With these premises, the researchers find that the statistics of defects follows a binomial distribution associated with the probability of forming a topological defect at the locations where multiple domains merge. They also find that all cumulants of the distribution are predicted to exhibit a common universal power-law scaling with the quench time in which the transition is crossed, in other words, this power law is fixed by the conventional KZM scaling. Most interestingly, this is a testable prediction.

These findings motivate the quest for universal signatures in the counting statistics of topological defects across the wide variety of experiments used to test KZM dynamics, leading to a better understanding of how order appears in different systems, including the early universe.

*Author: César Tomé López is a science writer and the editor of Mapping Ignorance*

*Disclaimer: Parts of this article may be copied verbatim or almost verbatim from the referenced research paper.*

## References

- Fernando J. Gómez-Ruiz, Jack J. Mayo, and Adolfo del Campo (2020) Full Counting Statistics of Topological Defects after Crossing a Phase Transition
*Physical Review Letters*doi: 10.1103/PhysRevLett.124.240602 ↩ - Smitha Vishveshwara (2020) Defect or No Defect: It’s a Toss Up
*Physics*13, 98 doi: 10.1103/Physics.13.98 ↩