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

We numerically study the phase structure of the CP(1) model in the presence of a topological $theta$-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 $0leq beta leq 1.1$ and find a CP-violating, first-order phase transition at $theta=pi$. In contrast to previous findings, our numerical results provide no evidence for a critical coupling $beta_c<1.1$ above which a second-order phase transition emerges at $theta=pi$ and/or the first-order transition line bifurcates at $theta eqpi$. If such a critical coupling exists, as suggested by Haldanes conjecture, our study indicates that is larger than $beta_c>1.1$.