Phase structure of the CP(1) model in the presence of a topological θ -term

Abstract

We numerically study the phase structure of the $CP(1)$ model in the presence of a topological $\ensuremath{\theta}$-term, a regime afflicted by the sign problem for conventional lattice Monte Carlo simulations. Using a bond-weighted tensor renormalization group method, we compute the free energy for inverse couplings ranging from $0\ensuremath{\le}\ensuremath{\beta}\ensuremath{\le}1.1$ and find a $CP$-violating, first-order phase transition at $\ensuremath{\theta}=\ensuremath{\pi}$. In contrast to previous findings, our numerical results provide no evidence for a critical coupling ${\ensuremath{\beta}}_{c}<1.1$ above which a second-order phase transition emerges at $\ensuremath{\theta}=\ensuremath{\pi}$ and/or the first-order transition line bifurcates at $\ensuremath{\theta}\ensuremath{\ne}\ensuremath{\pi}$. If such a critical coupling exists, as suggested by Haldane's conjecture, our study indicates that is larger than ${\ensuremath{\beta}}_{c}>1.1$.