1305.4695 (Wojciech H. Zurek)

Winding numbers and scaling tilts from random vortex-antivortex pairs    [PDF]

Wojciech H. Zurek

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.

View original: http://arxiv.org/abs/1305.4695