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

We numerically study the phase structure of the CP(1) model in the presence of a topological $θ$-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 $0\leq β\leq 1.1$ and find a CP-violating, first-order phase transition at $θ=π$. In contrast to previous findings, our numerical results provide no evidence for a critical coupling $β_c<1.1$ above which a second-order phase transition emerges at $θ=π$ and/or the first-order transition line bifurcates at $θ\neqπ$. If such a critical coupling exists, as suggested by Haldane&#39;s conjecture, our study indicates that is larger than $β_c>1.1$.