The matrix product state formalism is used to simulate Hamiltonian lattice gauge theories. To this end, we define matrix product state manifolds which are manifestly gauge invariant. As an application, we study 1+1 dimensional one flavor quantum electrodynamics, also known as the massive Schwinger model, and are able to determine very accurately the ground state properties and elementary one-particle excitations in the continuum limit. In particular, a novel particle excitation in the form of a heavy vector boson is uncovered, compatible with the strong coupling expansion in the continuum. We also study full quantum non-equilibrium dynamics by simulating the real-time evolution of the system induced by a quench in the form of a uniform background electric field. PACS numbers:Gauge theories hold a most prominent place in physics. They appear as effective low energy descriptions at different instances in condensed matter physics and nuclear physics. But far and foremost they lie at the root of our understanding of the four fundamental interactions that are each mediated by the gauge fields corresponding to a particular gauge symmetry. At the perturbative quantum level, this picture translates to the Feynman diagrammatic approach that has produced physical predictions with unlevelled precision, most famously in quantum electrodynamics (QED). However the perturbative approach miserably fails once the interactions become strong. This problem is most pressing for quantum chromodynamics (QCD), where all low energy features like quark confinement, chiral symmetry breaking and mass generation are essentially non-perturbative.Lattice QCD, which is based on Monte Carlo sampling of Wilson's Euclidean lattice version of gauge theories, has historically been by far the most successful method in tackling this strongly coupled regime. Using up a sizable fraction of the global supercomputer time, state of the art calculations have now reached impressive accuracy, for instance in the ab initio determination of the light hadron masses [1]. But in spite of its clear superiority, the lattice Monte Carlo sampling also suffers from a few drawbacks. There is the infamous sign problem that prevents application to systems with large fermionic densities. In addition, the use of Euclidean time, as opposed to real time, presents a serious barrier for the understanding of dynamical non-equilibrium phenomena. Over the last few years there has been a growing experimental and theoretical interest in precisely these elusive regimes, e.g. in the study of heavy ion collisions or early time cosmology.In this letter we study the application of tensor network states (TNS) as a possible complementary approach to the numerical simulation of gauge theories. This is highly relevant as this Hamiltonian method is free from the sign problem and allows for real-time dynamics. As a first application we focus on the massive Schwinger model. For this model the TNS approach has been studied before by Byrnes et al [2] and Bañuls et al [3]. By integrating out the ...

The formalism of matrix product states is used to perform a numerical study of (1+1)-dimensional QED -also known as the (massive) Schwinger model -in the presence of an external static 'quark' and 'antiquark'. We obtain a detailed picture of the transition from the confining state at short interquark distances to the broken-string 'hadronized' state at large distances and this for a wide range of couplings, recovering the predicted behavior in both the weak-and strong-coupling limit of the continuum theory. In addition to the relevant local observables like charge and electric field, we compute the (bipartite) entanglement entropy and show that subtraction of its vacuum value results in a UV-finite quantity. We find that both string formation and string breaking leave a clear imprint on the resulting entropy profile. Finally, we also study the case of fractional probe charges, simulating for the first time the phenomenon of partial string breaking.

It has been established that Matrix Product States can be used to compute the ground state and single-particle excitations and their properties of lattice gauge theories at the continuum limit. However, by construction, in this formalism the Hilbert space of the gauge fields is truncated to a finite number of irreducible representations of the gauge group. We investigate quantitatively the influence of the truncation of the infinite number of representations in the Schwinger model, oneflavour QED2, with a uniform electric background field. We compute the two-site reduced density matrix of the ground state and the weight of each of the representations. We find that this weight decays exponentially with the quadratic Casimir invariant of the representation which justifies the approach of truncating the Hilbert space of the gauge fields. Finally, we compute the single-particle spectrum of the model as a function of the electric background field.

Using Matrix Product Operators (MPO) the Schwinger model is simulated in thermal equilibrium. The variational manifold of gauge-invariant MPO is constructed to represent Gibbs states. As a first application, the chiral condensate in thermal equilibrium is computed and agreement with earlier studies is found. Furthermore, as a new application the Schwinger model is probed with a fractional charged static quark-antiquark pair separated infinitely far from each other. A critical temperature beyond which the string tension is exponentially suppressed is found, and is in qualitative agreement with analytical studies in the strong coupling limit. Finally, the CT symmetry breaking is investigated and our results strongly suggest that the symmetry is restored at any nonzero temperature.

Matrix Product States (MPS) are used for the simulation of the real-time dynamics induced by an electric quench on the vacuum state of the massive Schwinger model. For small quenches it is found that the obtained oscillatory behavior of local observables can be explained from the single-particle excitations of the quenched Hamiltonian. For large quenches damped oscillations are found and comparison of the late time behavior with the appropriate Gibbs states seems to give some evidence for the onset of thermalization. Finally, the MPS real-time simulations are compared with results from real-time lattice gauge theory which are expected to agree in the limit of large quenches.