  Over the last decade tensor network states (TNS) have emerged as a powerful
tool for the study of quantum many body systems. The matrix product states
(MPS) are one particular case of TNS and are used for the simulation of 1+1
dimensional systems. In [1] we considered the MPS formalism for the simulation
of the Hamiltonian lattice gauge formulation of 1+1 dimensional one flavor
quantum electrodynamics, also known as the massive Schwinger model. We deduced
the ground state and lowest lying excitations. Furthermore, we performed a full
quantum real-time simulation for a quench with a uniform background electric
field. In this proceeding we continue our work on the Schwinger model. We
demonstrate the advantage of working with gauge invariant MPS by comparing with
MPS simulations on the full Hilbert space, that includes numerous non-physical
gauge variant states. Furthermore, we compute the chiral condensate and recover
the predicted UV-divergent behavior.
