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

Abstract

The behavior of Wilson loops with fractional charge is used to study the $\theta$-dependence of the free energy density $\epsilon (\theta )$ for the $C{P}^1$, $C{P}^5$, and $C{P}^9$ 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 $C{P}^1$, $\epsilon (\theta )$ is smooth in the region $\theta \approx \pi$ and well-described by a dilute instanton gas throughout the range $0<\theta <2\pi$. For $C{P}^5$ and $C{P}^9$ the energy exhibits a clear cusp and evidence for discrete, degenerate vacua at $\theta =\pi$, as expected from large $N$ arguments. For $C{P}^9$ 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 $0<\theta <\pi$.

16 pages, 10 figures

References in corpus (2)

• Mass gap in the 2D O(3) non-linear sigma model with a theta=pi term
• Critical Behavior of CP^1 at theta = pi, Haldane's Conjecture and the Universality Class