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We investigate the $\theta$-dependence of 2-dimensional $CP^{N-1}$ models inthe large-$N$ limit by lattice simulations. Thanks to a recent algorithmproposed by M. Hasenbusch to improve the critical slowing down of topologicalmodes, combined with simulations at imaginary values of $\theta$, we manage todetermine 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 thenext-to-leading term in $1/N$ for the quadratic term in $\theta$ (topologicalsusceptibility), and up to the leading term for the quartic coefficient $b_2$.Moreover, we give a numerical estimate of further terms in the $1/N$ expansionfor both quantities, pointing out that the $1/N$ convergence for the$\theta$-dependence of this class of models is particularly slow.

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