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Entanglement of Gauge Theories: from the Toric Code to the Z2 Lattice Gauge Higgs Model

Wentao Xu, Michael Knap, Frank Pollmann

Published Oct 5, 2023• 0 citations
Publisher

Abstract

The scripts and data for the project are shown in three files. PEPS_contraction_lib.zip contains CTMRG, iTEBD methods for contracting the iPEPS, and also some exact diagonalization tools. PEPS_optim.zip contains code for optimizing the iPEPS, and Entanglement_spectrum_and_entropy.zip contains code for extraction of the entanglement spectrum and entropy. To use the code, one has to modify the file paths in the scripts.

DOI
10.5281/zenodo.8409970

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The scripts and data for the project are shown in three files. PEPS_contraction_lib.zip contains CTMRG, iTEBD methods for contracting the iPEPS, and also some exact diagonalization tools. PEPS_optim.zip contains code for optimizing the iPEPS, and Entanglement_spectrum_and_entropy.zip contains code for extraction of the entanglement spectrum and entropy. To use the code, one has to modify the file paths in the scripts.
This record contains code, notebooks, data, and figure scripts associated with the project. The description below lists the contents of the upload and their roles. strongsym.zip: Zip archive used for the Zenodo upload. It contains the simul/, finsize_data/, and dependencies/ directories. simul/ : Simulation and analysis notebooks and scripts. MF_*.ipynb - Mean-field analysis notebooks. finite_size_spins.ipynb - Finite-size dynamics for 2 level system alt_finite_size_spins.ipynb - Alternative finite-size with a stress on Liouvullian spectrum analysis in the two level system. liouv_spec_spin1.ipynb -Finite-size spectrum analysis for three level model. find_symmetry.ipynb - Symbolic symmetry-finding notebook. finsize_data/ : Data files used to generate figures. fig_script.ipynb: Notebook to generate the figures overview_sketch.ipynb: Notebook to generate schmatics and diagrams (no raw data) dependencies/ : Environment specification files, for both python (.yml) and julia (.toml) Usage: Unzip strongsym.zip, set up environments using the files in dependencies/, and run the notebooks in simul/. Data required for figures is in finsize_data/; figure scripts are at the repository root. For a more detailed walkthrough and documentation, see the GitHub repository: https://github.com/MarcNairn/spin_strong_symmetries_phase
A definition for the entanglement entropy in a gauge theory was given recently in arXiv:1501.02593. Working on a spatial lattice, it involves embedding the physical state in an extended Hilbert space obtained by taking the tensor product of the Hilbert space of states on each link of the lattice. This extended Hilbert space admits a tensor product decomposition by definition and allows a density matrix and entanglement entropy for the set of links of interest to be defined. Here, we continue the study of this extended Hilbert space definition with particular emphasis on the case of Non-Abelian gauge theories. We extend the electric centre definition of Casini, Huerta and Rosabal to the Non-Abelian case and find that it differs in an important term. We also find that the entanglement entropy does not agree with the maximum number of Bell pairs that can be extracted by the processes of entanglement distillation or dilution, and give protocols which achieve the maximum bound. Finally, we compute the topological entanglement entropy which follows from the extended Hilbert space definition and show that it correctly reproduces the total quantum dimension in a class of Toric code models based on Non-Abelian discrete groups.
Lattice gauge theories are a low-energy regularization of quantum field theories. They are the basis for the development of numerical algorithms that provide insight into the non-perturbative regime of quantum field theories which is difficult to achieve through other methods. We present a python package, ggpeps, implementing gauged Gaussian projected entangled pair states (GGPEPS), specialized to study lattice gauge theories. We give a an overview of the considered systems, describe the GGPEPS ansatz and provide a usage guide for the ggpeps package.