Meron- and Semi-Vortex-Clusters as Physical Carriers of Topological Charge and Vorticity

Bietenholz, Wolfgang; Pinto Barros, João C.; Caspar, Stephan; Hornung, Manes; Wiese, Uwe-Jens (27 August 2020). Meron- and Semi-Vortex-Clusters as Physical Carriers of Topological Charge and Vorticity. PoS - proceedings of science, LATTICE2019, p. 288. Trieste, Italy: Scuola Internazionale Superiore di Studi Avanzati SISSA 10.22323/1.363.0288

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In O(N) non-linear σ-models on the lattice, the Wolff cluster algorithm is based on rewriting the functional integral in terms of mutually independent clusters. Through improved estimators, the clusters are directly related to physical observables. In the (N − 1)-d O(N) model (with an appropriately constrained action) the clusters carry an integer or half-integer topological charge. Clusters with topological charge ±1/2 are denoted as merons. Similarly, in the 2-d O(2) model the clusters carry pairs of semi-vortices and semi-anti-vortices (with vorticity ±1/2) at their bound- ary. Using improved estimators, meron- and semi-vortex-clusters provide analytic insight into the topological features of the dynamics. We show that the histograms of the cluster-size distribu- tions scale in the continuum limit, with a fractal dimension D, which suggests that the clusters are physical objects. We demonstrate this property analytically for merons and non-merons in the 1-d O(2) model (where D = 1), and numerically for the 2-d O(2), 2-d O(3), and 3-d O(4) model, for which we observe fractal dimensions D < d. In the vicinity of a critical point, a scaling law relates D to a combination of critical exponents. In the 2-d O(3) model, meron- and multi-meron-clusters are responsible for a logarithmic ultraviolet divergence of the topological susceptibility.

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