Phase structure analysis of 2d lattice CP(1) model with $\theta$ term using tensor renormalization group method

We investigate the phase structure of a two-dimensional lattice CP(1) model with a $\theta$ term. In particular, we aim to identify a critical region expected to exist along a $\theta=\pi$ line. To explore the phase structure non-perturbatively and avoid the sign problem, we employ the tensor renormalization group method. We make two improvements compared to previous tensor network studies. The first improvement involves refining the initial tensor. Specifically, we construct it using a quadrature method, which achieves higher accuracy compared to the conventional approach. The second improvement consists of analyzing the phase structure using the information of the conformal field theory, namely the central charge and the scaling dimensions, which can be accessed relatively easily via the tensor renormalization group method. Thanks to these improvements, we identify both the onset of the critical region, $\beta_{\rm c}=0.5952(8)$ and its universality class as the SU(2)${}_{k=1}$ Wess-Zumino-Witten model.