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Simulations of lattice gauge theories with tensor networks and quantumcomputing have so far mainly focused on staggered fermions. In this paper, weuse matrix product states to study Wilson fermions in the Hamiltonianformulation and present a novel method to determine the additive massrenormalization. Focusing on the single-flavor Schwinger model as a benchmarkmodel, we investigate the regime of a nonvanishing topological $\theta$-term,which is inaccessible to conventional Monte Carlo methods. We systematicallyexplore the dependence of the mass shift on the volume, the lattice spacing,the $\theta$-parameter, and the Wilson parameter. This allows us to followlines of constant renormalized mass, and therefore to substantially improve thecontinuum extrapolation of the mass gap and the electric field density. Forsmall values of the mass, our continuum results agree with the theoreticalprediction from mass perturbation theory. Going beyond Wilson fermions, ourtechnique can also be applied to staggered fermions, and we demonstrate thatthe results of our approach agree with a recent theoretical prediction for themass shift at sufficiently large volumes.

Computing the Mass Shift of Wilson and Staggered Fermions in the Lattice Schwinger Model with Matrix Product States