  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. The
function $\epsilon(\theta)$ is extracted from the area law for a Wilson loop of
charge $q=\theta/2\pi$. For $CP^1$, $\epsilon(\theta)$ is smooth in the region
$\theta\approx\pi$ and well-described by a dilute instanton gas throughout the
range $0<\theta<2\pi$. For $CP^5$ and $CP^9$ the energy exhibits a clear cusp
and evidence for discrete, degenerate vacua at $\theta = \pi$, as expected from
large $N$ arguments. For $CP^9$ the $\theta$-dependence is in good quantitative
agreement with the leading order large $N$ prediction
$\epsilon(\theta)={1/2}\chi_t\theta^2$ throughout the range $0<\theta<\pi$.
