  Tensor network (TN) methods, in particular the Matrix Product States (MPS)
ansatz, have proven to be a useful tool in analyzing the properties of lattice
gauge theories. They allow for a very good precision, much better than standard
Monte Carlo (MC) techniques for the models that have been studied so far, due
to the possibility of reaching much smaller lattice spacings. The real reason
for the interest in the TN approach, however, is its ability, shown so far in
several condensed matter models, to deal with theories which exhibit the
notorious sign problem in MC simulations. This makes it prospective for dealing
with the non-zero chemical potential in QCD and other lattice gauge theories,
as well as with real-time simulations. In this paper, using matrix product
operators, we extend our analysis of the Schwinger model at zero temperature to
show the feasibility of this approach also at finite temperature. This is an
important step on the way to deal with the sign problem of QCD. We analyze in
detail the chiral symmetry breaking in the massless and massive cases and show
that the method works very well and gives good control over a broad range of
temperatures, essentially from zero to infinite temperature.
