Gauss Law, Minimal Coupling and Fermionic PEPS for Lattice Gauge Theories

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, by looking 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.

arXiv commentFourth version: minor revision of notes (third version) for SciPost. Notes originally prepared for two lectures given in the Focus week "Tensor Networks and Entanglement" of the workshop "Entanglement in Quantum System", at the Galileo Galilei Institute for Theoretical Physics (GGI), Florence, Italy in June 2018

References in corpus (45)

• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
• The phase diagram of dense QCD
• Real-time dynamics of lattice gauge theories with a few-qubit quantum computer
• Quantum Simulations of Lattice Gauge Theories using Ultracold Atoms in Optical Lattices
• Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks
• Ultracold Quantum Gases and Lattice Systems: Quantum Simulation of Lattice Gauge Theories
• Lattice gauge theories simulations in the quantum information era
• Fermionic Projected Entangled Pair States
• The mass spectrum of the Schwinger model with Matrix Product States
• Entanglement entropy in gauge theories and the holographic principle for electric strings
• A Formulation of Lattice Gauge Theories for Quantum Simulations
• Tensor networks for Lattice Gauge Theories and Atomic Quantum Simulation
• Density Matrix Renormalisation Group Approach to the Massive Schwinger Model
• Matrix product states for gauge field theories
• Real-time Dynamics in U(1) Lattice Gauge Theories with Tensor Networks
• Digital lattice gauge theories
• Gauging quantum states: from global to local symmetries in many-body systems
• The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems
• Tensor Networks for Lattice Gauge Theories with continuous groups
• Digital quantum simulation of ${\mathbb{Z}}_2$ lattice gauge theories with dynamical fermionic matter
• Digital quantum simulation of lattice gauge theories in three spatial dimensions
• Non-Abelian string breaking phenomena with Matrix Product States
• Density Induced Phase Transitions in the Schwinger Model: A Study with Matrix Product States
• Entanglement renormalization and gauge symmetry
• Lattice Gauge Tensor Networks
• Efficient Basis Formulation for (1+1)-Dimensional SU(2) Lattice Gauge Theory: Spectral calculations with matrix product states
• Thermal evolution of the Schwinger model with Matrix Product Operators
• Real-time simulation of the Schwinger effect with Matrix Product States
• Chiral condensate in the Schwinger model with Matrix Product Operators
• Hamiltonian simulation of the Schwinger model at finite temperature
• Finite-density phase diagram of a (1+1)-d non-abelian lattice gauge theory with tensor networks
• Removing Staggered Fermionic Matter in $U(N)$ and $SU(N)$ Lattice Gauge Theories
• Eliminating fermionic matter fields in lattice gauge theories
• Chiral projected entangled-pair state with topological order
• Fermionic Projected Entangled Pair States and Local U(1) Gauge Theories
• Building Projected Entangled Pair States with a Local Gauge Symmetry
• Projected Entangled Pair States with non-Abelian gauge symmetries: an SU(2) study
• Combining Tensor Networks with Monte Carlo Methods for Lattice Gauge Theories
• Phase Diagram and Conformal String Excitations of Square Ice using Gauge Invariant Matrix Product States
• Tensor Network Simulation of compact one-dimensional lattice Quantum Chromodynamics at finite density
• Matrix product representation of gauge invariant states in a Z_2 lattice gauge theory
• Towards overcoming the Monte Carlo sign problem with tensor networks
• Matrix product states and the nonabelian rotor model

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• Engineering a U(1) lattice gauge theory in classical electric circuits
• Finding the ground state of a lattice gauge theory with fermionic tensor networks: a $2+1d$ ${\mathbb{Z}}_2$ demonstration
• Selected topics of quantum computing for nuclear physics
• Duality as a Feasible Physical Transformation
• Local Manipulation and Measurement of Nonlocal Many-Body Operators in Lattice Gauge Theory Quantum Simulators
• Gauged Gaussian PEPS -- A High Dimensional Tensor Network Formulation for Lattice Gauge Theories
• Wilson Loops and Area Laws in Lattice Gauge Theory Tensor Networks
• Entanglement renormalization for gauge invariant quantum fields
• Lattice regularisation and entanglement structure of the Gross-Neveu model
• Finite-Entanglement Scaling of 2D Metals