Metadynamics Surfing on Topology Barriers: the $CP^{N-1}$ Case

TL;DR

This paper demonstrates that metadynamics can effectively overcome topological sector trapping in $CP^{N-1}$ models, enabling accurate computation of topological quantities crucial for understanding the $ heta$ vacuum and axion physics.

Contribution

It introduces the application of metadynamics to $CP^{N-1}$ models, allowing efficient reconstruction of free energy and topological susceptibility, with potential extension to QCD.

Findings

Successfully reconstructed the free energy $F(Q)$ for $CP^{20}$.

Computed topological susceptibility as a function of coupling and volume.

Showed potential for extending the method to QCD applications.

Abstract

As one approaches the continuum limit, $QCD$ systems, investigated via numerical simulations, remain trapped in sectors of field space with fixed topological charge. As a consequence the numerical studies of physical quantities may give biased results. The same is true in the case of two dimensional $CP^{N-1}$ models. In this paper we show that metadynamics, when used to simulate $CP^{N-1}$, allows to address efficiently this problem. By studying $CP^{20}$ we show that we are able to reconstruct the free energy of the topological charge $F(Q)$ and compute the topological susceptibility as a function of the coupling and of the volume. This is a very important physical quantity in studies of the dynamics of the $\theta$ vacuum and of the axion. This method can in principle be extended to $QCD$ applications.