Topological properties of $C{P}^{N-1}$ models in the large-$N$ limit

Abstract

We investigate, by numerical simulations on a lattice, the $\theta$-dependence of 2$d$ $C{P}^{N-1}$ models for a range of $N$ going from 9 to 31, combining imaginary $\theta$ and simulated tempering techniques to improve the signal-to-noise ratio and alleviate the critical slowing down of the topological modes. We provide continuum extrapolations for the second and fourth order coefficients in the Taylor expansion in $\theta$ of the vacuum energy of the theory, parameterized in terms of the topological susceptibility $\chi$ and of the so-called $b_2$ coefficient. Those are then compared with available analytic predictions obtained within the $1/N$ expansion, pointing out that higher order corrections might be relevant in the explored range of $N$, and that this fact might be related to the non-analytic behavior expected for $N=2$. We also consider sixth-order corrections in the $\theta$ expansion, parameterized in terms of the so-called $b_4$ coefficient: in this case our present statistical accuracy permits to have reliable non-zero continuum estimations only for $N\le 11$, while for larger values we can only set upper bounds. The sign and values obtained for $b_4$ are compared to large-$N$ predictions, as well as to results obtained for $SU(N_c)$ Yang-Mills theories, for which a first numerical determination is provided in this study for the case $N_c=2$.

14 pages, 17 eps figures

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• Large-$N$ expansion and $\theta$-dependence of $2d$ $C{P}^{N-1}$ models beyond the leading order
• $\theta$-dependence and center symmetry in Yang-Mills theories
• The Large $N_c$ limit of QCD on the lattice
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• $\theta$-dependence in the small-$N$ limit of $2d$ $C{P}^{N-1}$ models
• Large-$N$ ${\mathbb{CP}}^{N-1}$ sigma model on a Euclidean torus: uniqueness and stability of the vacuum
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