Applying Tensor Network Techniques to Lattice Gauge Theories

The term Tensor Network States (TNS) encloses a number of families that represent different ansatzes for the efficient description of the state of a quantum many-body system. The first of these families, Matrix Product States (MPS), lies at the basis of Density Matrix Renormalization Group methods, which have become the most precise tool for the study of one dimensional quantum many-body systems. Tensor Network Techniques reveal themselves as very promising tools for the non-perturbative study of lattice Hamiltonians. Specially suited to the study of equilibrium properties of local Hamiltonians, they can also be applied to non-local interactions and to dynamical problems. While the dimensions and sizes of the systems amenable to TNS studies are still far from those achievable by Lattice Gauge Theory computations, Tensor Networks can be readily used for problems which more standard techniques cannot easily tackle, such as the presence of a chemical potential, or out-of- equilibrium dynamics. As a proof of the feasibility of these methods, we have explored the performance of MPS techniques in the case of the Schwinger model, a widely used testbench for lattice techniques. The precision achieved by the method allows for accurate finite size and continuum limit extrapolations of the ground state energy, but also of the mass gaps, thus showing the feasibility of these techniques for gauge theory problems. Additionally, the MPS method can easily deal with the effect of finite temperature and chemical potential.