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

Abstract

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 $0\le \beta \le 1.1$ 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 $\theta \ne \pi$. If such a critical coupling exists, as suggested by Haldane's conjecture, our study indicates that is larger than ${\beta }_c>1.1$.

Comments: 10 pages, 9 figures, v2: updated to match journal version