Phase diagram near the quantum critical point in Schwinger model at $\theta = \pi$: analogy with quantum Ising chain

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.