Critical behavior of the lattice Schwinger model with a topological term at $\theta =\pi$ using the Grassmann tensor renormalization group

The lattice regularized Schwinger model with a so-called $\theta$ term is studied by using the Grassmann tensor renormalization group. We perform the Lee-Yang and Fisher zero analyses in order to investigate the phase structure at $\theta =\pi$. We find a first-order phase transition at larger fermion mass. Both the Lee-Yang zero and Fisher zero analyses indicate that the critical endpoint at which the first-order phase transition terminates belongs to the Ising universality class.

Figure 1

Expected phase diagram of Schwinger model with the $\theta$ term. The dotted line denotes a first-order phase transition, which terminates at a second-order phase transition point belonging to the Ising universality class.

Figure 2

Convergence of $\mathrm{Im}{\kappa }_0(L)$ for $L=128$ as a function of SVD truncation number $D$. Results for smaller $N_{\mathrm{ce}}$ and $N_{\mathrm{ce}}^{\prime }$ values are also presented at $D=160$. Note that they are plotted at slightly shifted values of $D$ in order to avoid the superposition of symbols and error bars.

Figure 3

Real (top) and imaginary (bottom) parts of the Fisher zero as a function of $L^{-1}$ at $\beta =10.0$. Solid curves represent the fit results with $\mathrm{Re}/\mathrm{Im}{\kappa }_0(L)=\mathrm{Re}/\mathrm{Im}{\kappa }_0(\infty )+a_{R/I}{L}^{-1/\nu }$ and dotted ones with Eqs. (27) and (28).

Figure 4

Imaginary part of the Lee-Yang zero for $\kappa =0.2400$ (top), $\kappa =0.2415$ (middle) and $\kappa =0.2430$ (bottom) as a function of $L^{-2}$ at $\beta =10.0$. Solid curves represent the fit results with $\mathrm{Im}{\theta }_0(L)=\mathrm{Im}{\theta }_0(\infty )+a{L}^{-y}$. Intercepts are magnified in small windows.