title: 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 W trapped inside a loop of circumference C with the square root of that circumference, W ≈√C, when the expected winding numbers are large, |W| ≫ 1. Such scaling is consistent with the Kibble-Zurek mechanism (KZM), with <W2> inversely proportional to ξ, the typical size of the domain that can break symmetry in unison. (The dependence of ξ on quench rate is predicted by KZM from critical exponents of the phase transition.) Thus, according to KZM, the dispersion √<W2> scales as √C/ξ for large W. 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 C (i.e., ≈√<W2> ≈C). Scaling of the dispersion of W as well as of the probability of detection of non-zero W with C and ξ can be also studied for loops so small that non-zero windings are rare. In this case I show that dispersion varies not as 1/√ξ, but as 1/ξ, which results in a doubling of the scaling of dispersion with the quench rate when compared to the large |W| regime. Moreover, the probability of trapping of non-zero W becomes approximately equal to <W2>, and scales as 1/ξ2. This quadruples--as compared with √<W2>≃ √C/ ξ valid for large W--the exponent in the power law dependence of the frequency of trapping of |W| = 1 on ξ when the probability of |W| > 1 is negligible. This change of the power law exponent by a factor of four--from 1√ξ for the dispersion of large W to 1/ξ2 for the frequency of non-zero W when |W| > 1 is negligibly rare--is of paramount importance for experimental tests of KZM.
authors.affiliation: Theory Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.
authors.forename: W H
authors.initials: WH
authors.lastname: Zurek
mesh_terms: D003198:Computer Simulation; D008956:Models, Chemical; D008958:Models, Molecular; D015233:Models, Statistical; D044367:Phase Transition