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The behavior of Wilson loops with fractional charge is used to study the$\theta$-dependence of the free energy density $\epsilon(\theta)$ for the$CP^1$, $CP^5$, and $CP^9$ sigma models in two spacetime dimensions. Thefunction $\epsilon(\theta)$ is extracted from the area law for a Wilson loop ofcharge $q=\theta/2\pi$. For $CP^1$, $\epsilon(\theta)$ is smooth in the region$\theta\approx\pi$ and well-described by a dilute instanton gas throughout therange $0<\theta<2\pi$. For $CP^5$ and $CP^9$ the energy exhibits a clear cuspand evidence for discrete, degenerate vacua at $\theta = \pi$, as expected fromlarge $N$ arguments. For $CP^9$ the $\theta$-dependence is in good quantitativeagreement with the leading order large $N$ prediction$\epsilon(\theta)={1/2}\chi_t\theta^2$ throughout the range $0<\theta<\pi$.

Fractionally charged Wilson loops as a probe of $\theta$-dependence in $CP^{N-1}$ sigma models: Instantons vs. large N