Topological Charge Distribution and $CP^1$ Model with $\theta$ Term

Abstract

The two dimensional $CP^1$ model with $\theta$ term is simulated. We compute the topological charge distribution $P(Q)$ by employing the ``set method" and ``trial function method", which are effective in the calculations for very wide range of $Q$ and large volume. The distribution $P(Q)$ shows the Gaussian behavior in the small $\beta$ (inverse coupling constant) region and deviates from it in the large $\beta$ region. The free energy and its moment are calculated as a function of $\theta$. For small $\beta$, the partition function is given by the elliptic theta function, and the distribution of its zeros on the complex $\theta$ plane leads to the first order phase transition at $\theta=\pi$. In the large $\beta$ region, on the other hand, this first order phase transition disappears, but definite conclusion concerning the transition is not reached due to large errors.