  Lattice computations in the Hamiltonian formulation have so far mainly
focused on staggered fermions. In these proceedings, we study Wilson fermions
in the Hamiltonian formulation and propose a new method to determine the
resulting mass shift. As a benchmark study, we examine the one-flavour
Schwinger model with Wilson fermions and a topological $\theta$-term using
matrix product states. Wilson fermions explicitly break chiral symmetry; thus,
the bare mass of the lattice model receives an additive renormalization. In
order to measure this mass shift directly, we develop a method that is suitable
for the Hamiltonian formulation, which relies on the fact that the vacuum
expectation value of the electric field density vanishes when the renormalized
mass is zero. We examine the dependence of the mass shift on the lattice
spacing, the lattice volume, the $\theta$-parameter, and the Wilson parameter.
Using the mass shift, we then perform the continuum extrapolation of the
electric field density and compare the resulting mass dependence to the
analytical predictions of mass perturbation theory. We demonstrate that
incorporating the mass shift significantly improves the continuum
extrapolation. Finally, we apply our method to the same model using staggered
fermions instead of Wilson fermions and compare the resulting mass shift to
recent theoretical predictions.
