Meron-Cluster Simulation of the $\theta$-Vacuum in the 2-d $O(3)$-Model

TL;DR

This paper introduces a cluster algorithm for simulating the 2D $O(3)$-model with a $ heta$-vacuum, revealing a second order phase transition at $ heta = \pi$ and confirming theoretical predictions.

Contribution

It develops a Meron-Cluster simulation method for the $ heta$-vacuum in the 2D $O(3)$-model, enabling numerical study of phase transitions and topological effects.

Findings

Critical exponents match the $k=1$ Wess-Zumino-Novikov-Witten model.

Merons are identified as half-integer topological charge clusters.

Results support Haldane's conjecture for 1D antiferromagnetic chains.

Abstract

The 2-d $O(3)$-model with a $\theta$-vacuum term is formulated in terms of Wolff clusters. Each cluster carries a half-integer topological charge. The clusters with charge $\pm 1/2$ are identified as merons. At $\theta = \pi$ the merons are bound in pairs inducing a second order phase transition at which the mass-gap vanishes. The construction of an improved estimator for the topological charge distribution makes numerical simulations of the phase transition feasible. The measured critical exponents agree with those of the $k = 1$ Wess-Zumino-Novikov-Witten model. Our results are consistent with Haldane's conjecture for 1-d antiferromagnetic quantum spin chains.