Topological relics of symmetry breaking: Winding numbers and scaling tilts from random vortex-antivortex pairs

TL;DR

This paper demonstrates that the distribution of vortex-antivortex pairs leads to specific scaling laws for winding numbers in topological defects, providing insights into the Kibble-Zurek mechanism and experimental detection.

Contribution

It introduces a novel analysis of vortex-antivortex pair distributions and their impact on winding number scaling, contrasting with individual vortex models.

Findings

Winding number W scales with square root of circumference C for vortex-antivortex pairs.

In small loops, the dispersion of W scales with area, quadrupling the exponent compared to large W regimes.

Probability of trapping non-zero W scales with the area, significantly affecting the frequency of winding number trapping.

Abstract

I show that random distributions of vortex-antivortex pairs (rather than of individual vortices) lead to scaling of typical winding numbers W trapped inside a loop of circumference C with the square root of C when the expected winding numbers are large. Such scaling is consistent with the Kibble-Zurek mechanism (KZM). By contrast, distribution of individual vortices with randomly assigned topological charges would result in the dispersion of W scaling with the square root of the area inside C. Scaling of the dispersion of W and of the probability of detection of non-zero W with C can be also studied for loops so small that non-zero windings are rare. In this case I show a doubling of the scaling of dispersion with C when compared to the scaling of dispersion in the large W regime. Moreover, probability of trapping of a non-zero W becomes, in this case, proportional to the area subtended by C (hence, to the square of circumference). This quadruples, as compared with large winding numbers regime, the exponent in the power law dependence of the frequency of trapping of W=+1 or W=-1 on C. Such change of the power law exponent by a FACTOR OF FOUR implies quadrupling of the scaling of the frequency of winding number trapping with the quench rate, and is of key importance for experimental tests of KZM.