  We numerically study the single-flavor Schwinger model with a topological
$\theta$-term, which is practically inaccessible by standard lattice Monte
Carlo simulations due to the sign problem. By using numerical methods based on
tensor networks, especially the one-dimensional matrix product states, we
explore the non-trivial $\theta$-dependence of several lattice and continuum
quantities in the Hamiltonian formulation. In particular, we compute the
ground-state energy, the electric field, the chiral fermion condensate, and the
topological vacuum susceptibility for positive, zero, and even negative fermion
mass. In the chiral limit, we demonstrate that the continuum model becomes
independent of the vacuum angle $\theta$, thus respecting CP invariance, while
lattice artifacts still depend on $\theta$. We also confirm that negative
masses can be mapped to positive masses by shifting $\theta\rightarrow \theta
+\pi$ due to the axial anomaly in the continuum, while lattice artifacts
non-trivially distort this mapping. This mass regime is particularly
interesting for the (3+1)-dimensional QCD analog of the Schwinger model, the
sign problem of which requires the development and testing of new numerical
techniques beyond the conventional Monte Carlo approach.
