Phase structure of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>C</mml:mi><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> model in the presence of a topological <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>θ</mml:mi></mml:math> -term

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