Schwinger model on an interval: analytic results and DMRG

Quantum electrodynamics in $1+1$ dimensions (Schwinger model) on an interval admits lattice discretization with a finite-dimensional Hilbert space, and is often used as a testbed for quantum and tensor network simulations. In this work we clarify the precise mapping between the boundary conditions in the continuum and lattice theories. In particular we show that the conventional Gauss law constraint commonly used in simulations induces a strong boundary effect on the charge density, reflecting the appearance of fractionalized charges. Further, we obtain by bosonization a number of exact analytic results for local observables in the massless Schwinger model. We compare these analytic results with the simulation results obtained by the density matrix renormalization group (DMRG) method and find excellent agreements.