Topological relics of symmetry breaking: winding numbers and scaling tilts from random vortex–antivortex pairs

Abstract

I show that random distributions of vortex–antivortex pairs (rather than of individual vortices) lead to scaling of typical winding numbers \U0001d4b2 trapped inside a loop of circumference \U0001d49e with the square root of that circumference, , when the expected winding numbers are large, |\U0001d4b2| ≫ 1. Such scaling is consistent with the Kibble–Zurek mechanism (KZM), with 〈\U0001d4b22〉 inversely proportional to , the typical size of the domain that can break symmetry in unison. (The dependence of on quench rate is predicted by KZM from critical exponents of the phase transition.) Thus, according to KZM, the dispersion scales as for large \U0001d4b2. By contrast, a distribution of individual vortices with randomly assigned topological charges would result in the dispersion scaling with the square root of the area inside \U0001d49e (i.e., ). Scaling of the dispersion of \U0001d4b2 as well as of the probability of detection of non-zero \U0001d4b2 with \U0001d49e and can be also studied for loops so small that non-zero windings are rare. In this case I show that dispersion varies not as , but as , which results in a doubling of the scaling of dispersion with the quench rate when compared to the large |\U0001d4b2| regime. Moreover, the probability of trapping of non-zero \U0001d4b2 becomes approximately equal to 〈\U0001d4b22〉, and scales as . This quadruples—as compared with valid for large \U0001d4b2—the exponent in the power law dependence of the frequency of trapping of |\U0001d4b2| = 1 on when the probability of |\U0001d4b2| > 1 is negligible. This change of the power law exponent by a factor of four—from for the dispersion of large \U0001d4b2 to for the frequency of non-zero \U0001d4b2 when |\U0001d4b2| > 1 is negligibly rare—is of paramount importance for experimental tests of KZM.