Topological Charge Distribution and $CP^1$ Model with $θ$ Term

Abstract

The two dimensional $CP^1$ model with $θ$ 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 $β$ (inverse coupling constant) region and deviates from it in the large $β$ region. The free energy and its moment are calculated as a function of $θ$. For small $β$, the partition function is given by the elliptic theta function, and the distribution of its zeros on the complex $θ$ plane leads to the first order phase transition at $θ=π$. In the large $β$ region, on the other hand, this first order phase transition disappears, but definite conclusion concerning the transition is not reached due to large errors.