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

TL;DR

This paper investigates the topological properties of clusters in O(N) models, showing they carry quantized topological charges and exhibit fractal scaling, providing physical insight into their role in critical phenomena.

Contribution

It introduces the concept of meron- and semi-vortex-clusters as physical carriers of topological charge, with analytical and numerical evidence of their fractal nature across various models.

Findings

Clusters carry quantized topological charge.

Cluster size distributions scale with a fractal dimension.

Meron clusters influence topological susceptibility divergence.

Abstract

In O($N$) non-linear $\sigma$-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 $\pm 1/2$ 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 $\pm 1/2$) at their boundary. 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 distributions 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.