Large-$N$ expansion and $\theta$-dependence of $2d$ $CP^{N-1}$ models beyond the leading order

TL;DR

This study uses advanced lattice simulations to analyze the $ heta$-dependence of 2D $CP^{N-1}$ models at large N, confirming analytic predictions and exploring higher-order terms with improved computational methods.

Contribution

It provides the first detailed numerical determination of higher-order $ heta$-dependence terms beyond the leading order in large-$N$ $CP^{N-1}$ models using improved algorithms.

Findings

Results support analytic predictions for topological susceptibility and $b_2$ coefficients.

Numerical estimates of higher-order terms in $1/N$ expansion.

Convergence in $1/N$ for $ heta$-dependence is slow.

Abstract

We investigate the $\theta$-dependence of 2-dimensional $CP^{N-1}$ models in the large-$N$ limit by lattice simulations. Thanks to a recent algorithm proposed by M. Hasenbusch to improve the critical slowing down of topological modes, combined with simulations at imaginary values of $\theta$, we manage to determine the vacuum energy density up the sixth order in $\theta$ and up to $N = 51$. Our results support analytic predictions, which are known up to the next-to-leading term in $1/N$ for the quadratic term in $\theta$ (topological susceptibility), and up to the leading term for the quartic coefficient $b_2$. Moreover, we give a numerical estimate of further terms in the $1/N$ expansion for both quantities, pointing out that the $1/N$ convergence for the $\theta$-dependence of this class of models is particularly slow.