# Testing Haldane's Conjecture in the O(3) Model by a Meron Cluster Simulation

## TL;DR

This paper uses a cluster algorithm to simulate the 2D O(3) model with a  term, confirming Haldane's conjecture of a phase transition at  = , consistent with the WZNW model predictions.

## Contribution

The study introduces a meron cluster simulation method for the 2D O(3) model with a  term, providing reliable evidence for the phase transition at  = .

## Key findings

- Critical exponents match WZNW model predictions.
- Mass gap vanishes at  = , indicating a second order phase transition.
- Meron clusters effectively identify topological charge configurations.

## Abstract

Twelve years ago, Haldane formulated his famous conjecture for 1-d antiferromagnetic quantum spin chains. In the context of the 2-d O(3) model with a \theta term, it predicts a phase transition at \theta = \pi, which has not yet been verified reliably. To simulate this we use the Wolff cluster algorithm together with an improved estimator for the topological charge distribution. Each cluster carries integer or half integer charge. Clusters with charge 1/2 are identified with merons. At \theta = \pi they are inactive, such that the mass gap vanishes. We obtain critical exponents which are consistent with predictions from the k=1 WZNW model, therefore confirming a second order phase transition.

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Source: https://tomesphere.com/paper/hep-lat/9509053