  The Schwinger model, one-dimensional quantum electrodynamics, has CP symmetry
at $\theta = \pi$ due to the topological nature of the $\theta$ term. At zero
temperature, it is known that as increasing the fermion mass, the system
undergoes a second-order phase transition to the CP broken phase, which belongs
to the same universality class as the quantum Ising chain. In this paper, we
obtain the phase diagram near the quantum critical point (QCP) in the
temperature and fermion mass plane using first-principle Monte Carlo
simulations, while avoiding the sign problem by using the lattice formulation
of the bosonized Schwinger model. Specifically, we perform a detailed
investigation of the correlation function of the electric field near the QCP
and find that its asymptotic behavior can be described by the universal scaling
function of the quantum Ising chain. This finding indicates the existence of
three regions near the QCP, each characterized by a specific asymptotic form of
the correlation length, and demonstrates that the CP symmetry is restored at
any nonzero temperature, entirely analogous to the quantum Ising chain. The
range of the scaling behavior is also examined and found to be particularly
wide.
