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arXiv:2412.01737 · Journal of High Energy Physics

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

Gertian Roose, Erez Zohar

Published Nov 11, 2025• 1 citation

Abstract

A bstract 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.

DOI: 10.1007/jhep11(2025)052

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A bstract 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.

Tensor network states, and in particular projected entangled pair states (PEPS) have been a strong ansatz for the variational study of complicated quantum many-body systems, thanks to their built-in entanglement entropy area law. In this work, we use a special kind of PEPS---gauged Gaussian fermionic PEPS (GGFPEPS)---to find the ground state of

2+1

dimensional pure

{\mathbb{Z}}_{2}

lattice gauge theories for a wide range of coupling constants. We do so by combining PEPS methods with Monte-Carlo computations, allowing for efficient contraction of the PEPS and computation of correlation functions. Previously, such numerical computations involved the calculation of the Pfaffian of a matrix scaling with the system size, forming a severe bottleneck; in this work we show how to overcome this problem. This paves the way for applying the method we propose and benchmark here to other gauge groups, higher dimensions, and models with fermionic matter, in an efficient, sign-problem-free way.

Lattice gauge theories (LGTs) provide a powerful framework for studying non-perturbative phenomena in gauge theories. However, conventional approaches such as Monte Carlo (MC) simulations in imaginary time are limited, as they do not allow real time evolution and suffer from a sign problem in many important cases. Using Gauged Gaussian fermionic projected entangled pair states (GGFPEPS) as a variational ground state ansatz offers an alternative for studying LGTs through a sign-problem-free variational MC. As this method is extended to larger and more complex systems, understanding its numerical behavior becomes essential. While conventional action based MC has been extensively studied, the performance and characteristics of non-action-based MC within the GGFPEPS framework are far less explored. In this work, we investigate these algorithmic aspects, identifying an optimal update size for GGFPEPS-based MC simulations for

\mathbb{Z}_2

in

2+1

dimensions. We show that gauge fixing generally slows convergence, and demonstrate that not exploiting the translation-invariance can, in some cases, improve the computational time scaling of error convergence. We expect that these improvements will allow advancing the simulation to larger and more complex systems.

Gauge theories form the basis of our understanding of modern physics—ranging from the description of quarks and gluons to effective models in condensed matter physics. In the nonperturbative regime, gauge theories are conventionally treated discretely as lattice gauge theories. The resulting systems are evaluated with path-integral based Monte Carlo methods. These methods, however, can suffer from the sign problem and do not allow for a direct evaluation of real-time dynamics. In this work, we present a unified and comprehensive framework for gauged Gaussian projected entangled pair states (PEPS), a variational ansatz based on tensor networks. We review the construction of Hamiltonian lattice gauge theories, explain their similarities to PEPS, and detail the construction of the ansatz state. The estimation of ground states is based on a variational Monte Carlo procedure with the PEPS as an ansatz state. This sign-problem-free ansatz can be efficiently evaluated in any dimension with arbitrary gauge groups, and can include dynamical fermionic matter, suggesting new options for the simulation of nonperturbative regimes of gauge theories, including quantum chromodynamics. Published by the American Physical Society 2024

Fermionic Gaussian projected entangled pair states (PEPS) are fermionic tensor network state constructions that describe the physics of ground states of noninteracting fermionic Hamiltonians. As noninteracting states, one may study and analyze them very efficiently, in both analytical and numerical means. Recently it was shown that they may be used as the starting point---after applying so-called PEPS gauging mechanisms---for variational study of nontrivial, interacting states of lattice gauge theories. This is done using sign-problem free variational Monte Carlo techniques. In this work we show how to generalize such states from two to three spatial dimensions, focusing on spin representations and requirements of lattice rotations. We present constructions that are crucial for the application of the above-mentioned variational Monte Carlo techniques for studying nonperturbative lattice gauge theory physics, with fermionic matter, in

2+1\mathrm{D}

and

3+1\mathrm{D}

models. Thus, the constructions presented here are crucial for the study of nontrivial lattice gauge theories with fermionic tensor network states.