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arXiv:1807.01294 · SciPost Physics Lecture Notes

Gauss law, minimal coupling and fermionic PEPS for lattice gauge theories

Patrick Emonts, Erez Zohar

Published Jan 17, 2020• 21 citations
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Abstract

In these lecture notes, we review some recent works on Hamiltonian lattice gauge theories, that involve, in particular, tensor network methods. The results reviewed here are tailored together in a slightly different way from the one used in the contexts where they were first introduced. We look at the Gauss law from two different points of view: for the gauge field, it is a differential equation, while from the matter point of view, on the other hand, it is a simple, explicit algebraic equation. We will review and discuss what these two points of view allow and do not allow us to do, in terms of unitarily gauging a pure-matter theory and eliminating the matter from a gauge theory, and relate that to the construction of PEPS (Projected Entangled Pair States) for lattice gauge theories.

DOI
10.21468/scipostphyslectnotes.12

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In these lecture notes, we review some recent works on Hamiltonian lattice gauge theories, that involve, in particular, tensor network methods. The results reviewed here are tailored together in a slightly different way from the one used in the contexts where they were first introduced. We look at the Gauss law from two different points of view: for the gauge field, it is a differential equation, while from the matter point of view, on the other hand, it is a simple, explicit algebraic equation. We will review and discuss what these two points of view allow and do not allow us to do, in terms of unitarily gauging a pure-matter theory and eliminating the matter from a gauge theory, and relate that to the construction of PEPS (Projected Entangled Pair States) for lattice gauge theories.
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+12+12+1 dimensional pure Z2{\mathbb{Z}}_{2}Z2​ 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.
Tensor network states, and in particular projected entangled pair states (PEPS), suggest an innovative approach for the study of lattice gauge theories, both from a pure theoretic point of view, and as a tool for the analysis of the recent proposals for quantum simulations of lattice gauge theories.In this paper we present a framework for describing locally gauge invariant states on lattices using PEPS.The PEPS constructed hereby shall include both bosonic and fermionic states, suitable for all combinations of matter and gauge fields in lattice gauge theories defined by either finite or compact Lie groups.
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+1D2+1\mathrm{D}2+1D and 3+1D3+1\mathrm{D}3+1D models. Thus, the constructions presented here are crucial for the study of nontrivial lattice gauge theories with fermionic tensor network states.