Gauging a superposition of fermionic Gaussian projected entangled pair states to get lattice gauge theory eigenstates

TL;DR

The work develops gauged superpositions of Gaussian PEPS (GSGFPEPS) as a robust, sign-problem-free framework for simulating lattice gauge theories by integrating Monte Carlo sampling over gauge backgrounds with Gaussian tensor-network contractions. It proves that any LGT ground state admits an exact representation as a gauged PEPS that is non-Gaussian only on bonds, and argues that a small number of Gaussian components suffices to capture bound states on top of the strongly coupled vacuum, enabling efficient computation of gauge-invariant observables. By separating the construction into a pure gauge part, a fermionic part, and a past-gauge-elimination step, the authors derive exact representations and practical pathways to variational optimization, with the potential to reach continuum limits via gauged continuous tensor networks. The approach has direct relevance for studying bound-state phenomena such as hadrons and mesons in QCD-like theories and offers a promising route to sign-problem-free, real-time explorations of lattice gauge dynamics.

Abstract

Gauged fermionic projected entangled pair states (GFPEPS) and their Gaussian counterpart (GGFPEPS) are a novel type of lattice gauge theory Ansatz state that combine ideas from the Monte Carlo and tensor network communities. In particular, computation of observables for such states boils down to a Monte Carlo integration over possible gauge field configurations that have probabilities dictated by a fermionic tensor network contraction that accounts for the matter in that background configuration. Crucially, this probability distribution is positive definite and real so that there is no sign problem. When the underlying PEPS is Gaussian, tensor network contraction can be done efficiently, and in this scenario the Ansatz has been tested well numerically. In this work we propose to gauge superpositions of Gaussian PEPS and demonstrate that one can still efficiently compute observables when few Gaussians are in the superposition. As we will argue, the latter is exactly the case for bound states on top of the strongly interacting LGT vacuum, which makes this Ansatz particularly suitable for that scenario. As a corollary, we will provide an exact representation of the LGT ground state as a gauged PEPS.