  The study of lattice gauge theories with Monte Carlo simulations is hindered
by the infamous sign problem that appears under certain circumstances, in
particular at non-zero chemical potential. So far, there is no universal method
to overcome this problem. However, recent years brought a new class of
non-perturbative Hamiltonian techniques named tensor networks, where the sign
problem is absent. In previous work, we have demonstrated that this approach,
in particular matrix product states in 1+1 dimensions, can be used to perform
precise calculations in a lattice gauge theory, the massless and massive
Schwinger model. We have computed the mass spectrum of this theory, its thermal
properties and real-time dynamics. In this work, we review these results and we
extend our calculations to the case of two flavours and non-zero chemical
potential. We are able to reliably reproduce known analytical results for this
model, thus demonstrating that tensor networks can tackle the sign problem of a
lattice gauge theory at finite density.
