Topological susceptibility of $2d~\mathrm{CP}^1$ or $\mathrm{O}(3)$ non-linear $\sigma$-model: is it divergent or not?

TL;DR

This paper investigates whether the topological susceptibility of the 2D CP^1 (O(3)) non-linear sigma model diverges, using non-perturbative lattice simulations and a novel fixed-volume approach, confirming the divergence.

Contribution

The study introduces a fixed-volume lattice approach and a multicanonic algorithm to non-perturbatively confirm the divergence of topological susceptibility in the 2D CP^1 model.

Findings

Confirmed divergence of topological susceptibility in 2D CP^1 model

Validated fixed-volume approach for studying topological properties

Demonstrated effectiveness of multicanonic algorithm in rare fluctuation sampling

Abstract

The topological susceptibility of $2d$ $\mathrm{CP}^{N-1}$ models is expected, based on perturbative computations, to develop a divergence in the limit $N \to 2$, where these models reduce to the well-known non-linear $\mathrm{O}(3)$ $\sigma$-model. The divergence is due to the dominance of instantons of arbitrarily small size and its detection by numerical lattice simulations is notoriously difficult, because it is logarithmic in the lattice spacing. We approach the problem from a different perspective, studying the behavior of the model when the volume is fixed in dimensionless lattice units, where perturbative predictions are turned into more easily checkable behaviors. After testing this strategy for $N = 3$ and $4$, we apply it to $N = 2$, adopting at the same time a multicanonic algorithm to overcome the problem of rare topological fluctuations on asymptotically small lattices. Our final results fully confirm, by means of purely non-perturbative methods, the divergence of the topological susceptibility of the $2d$ $\mathrm{CP}^1$ model.