Katsumasa NakayamaLena FunckeKarl JansenYING-JER KAOStefan Kühn2023-07-212023-07-212021-07-3024700010https://scholars.lib.ntu.edu.tw/handle/123456789/63401710 pages, 9 figures, v2: updated to match journal versionWe 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 $0\leq \beta \leq 1.1$ 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 $\theta\neq\pi$. If such a critical coupling exists, as suggested by Haldane's conjecture, our study indicates that is larger than $\beta_c>1.1$.High Energy Physics - Lattice; High Energy Physics - Lattice; Physics - Strongly Correlated Electrons; Quantum Physics[SDGs]SDG7journal article10.1103/PhysRevD.105.0545072-s2.0-85126961366http://arxiv.org/abs/2107.14220v2