$\theta$-dependence in the small-$N$ limit of $2d$ $C{P}^{N-1}$ models

Abstract

We present a systematic numerical study of $\theta$-dependence around $\theta =0$ in the small-$N$ limit of $2d$ $C{P}^{N-1}$ models, aimed at clarifying the possible presence of a divergent topological susceptibility in the continuum limit. We follow a twofold strategy, based on one side on direct simulations for $N=2$ and $N=3$ on lattices with correlation lengths up to $O(10^2)$, and on the other side on the small-$N$ extrapolation of results obtained for $N$ up to $9$. Based on that, we provide conclusive evidence for a finite topological susceptibility at $N=3$, with a continuum estimate ${\xi }^2\chi =0.110(5)$. On the other hand, results obtained for $N=2$ are still inconclusive: they are consistent with a logarithmically divergent continuum extrapolation, but do not yet exclude a finite continuum value, ${\xi }^2\chi \sim 0.4$, with the divergence taking place for $N$ slightly below 2 in this case. Finally, results obtained for the non-quadratic part of $\theta$-dependence, in particular for the so-called $b_2$ coefficient, are consistent with a $\theta$-dependence matching that of the Dilute Instanton Gas Approximation at the point where ${\xi }^2\chi$ diverges.

15 pages, 17 eps figures, minor changes

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• $\theta$ dependence of 4D $SU(N)$ gauge theories in the large-$N$ limit
• Topological Lattice Actions
• Fighting topological freezing in the two-dimensional CP${}^{N-1}$ model
• Mass gap in the 2D O(3) non-linear sigma model with a theta=pi term
• Large-$N$ expansion and $\theta$-dependence of $2d$ $C{P}^{N-1}$ models beyond the leading order
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• On the effective Lagrangian of CP^(N-1) models in the large N limit
• Small Instantons in $C{P}^1$ and $C{P}^2$ Sigma Models

Cited by in corpus (5)

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