Wilson Loops and Area Laws in Lattice Gauge Theory Tensor Networks

Abstract: Tensor network states have been a very prominent tool for the study of quantum many-body physics, thanks to their physically relevant entanglement properties and their ability to encode symmetries. In the last few years, the formalism has been extended and applied to theories with local symmetries to - lattice gauge theories. In the contraction of tensor network states as well as correlation functions of physical observables with respect to them, one uses the so-called transfer operator, whose local properties dictate the long-range behaviour of the state. In this work we study transfer operators of tensor network states (in particular, PEPS - projected entangled pair states) in the context of lattice gauge theories, and consider the implications of the local symmetry on their structure and properties. We focus on the Wilson loop - a nonlocal, gauge-invariant observable which is central to pure gauge theories, whose long range decay behaviour probes the confinement or deconfinement of static charges. Using the symmetry, we show how to handle its contraction, and formulate conditions relating local properties to its decay fashion.