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States of 2D Yang-Mills and Large-Volume Entanglement

Dmitry Melnikov, Jefferson T. Oliveira, Valmir Peixoto, Marcia Tenser

Abstract

We study entanglement in two-dimensional Yang-Mills theory, viewed as a quasi-topological model of emergent space. The most familiar class of states in this theory are states defined by Euclidean path integrals over Riemann surfaces. Bipartite states of this class have thermofield double structure, with entanglement consistently reducing with total area and the number of topological defects, turning separable in the infinite-area limit. In contrast, Wilson lines and loops generate rich non-monotonic behavior of the entanglement entropy. Most notably, we find that for a certain discrete set of configurations, entanglement remains finite at infinite area. The reduced density matrices, in such configurations, take the form of finite-dimensional projectors onto non-trivial vacuum sectors. We also discuss the implications of the large-volume effects for confinement and find that special asymptotic configurations are related to transitions in the confining force.

States of 2D Yang-Mills and Large-Volume Entanglement

Abstract

We study entanglement in two-dimensional Yang-Mills theory, viewed as a quasi-topological model of emergent space. The most familiar class of states in this theory are states defined by Euclidean path integrals over Riemann surfaces. Bipartite states of this class have thermofield double structure, with entanglement consistently reducing with total area and the number of topological defects, turning separable in the infinite-area limit. In contrast, Wilson lines and loops generate rich non-monotonic behavior of the entanglement entropy. Most notably, we find that for a certain discrete set of configurations, entanglement remains finite at infinite area. The reduced density matrices, in such configurations, take the form of finite-dimensional projectors onto non-trivial vacuum sectors. We also discuss the implications of the large-volume effects for confinement and find that special asymptotic configurations are related to transitions in the confining force.
Paper Structure (31 sections, 136 equations, 28 figures)

This paper contains 31 sections, 136 equations, 28 figures.

Table of Contents

  1. Introduction
  2. States in pure 2D Yang-Mills theory
  3. The disk
  4. The cylinder
  5. Higher genera
  6. Entanglement entropy analysis
  7. Configurations with Wilson loops
  8. Contractible loops without intersections
  9. Entanglement entropy analysis for a contractible loop
  10. Entanglement entropy analysis for multiple contractible loops
  11. Contractible loops with intersections
  12. Entanglement entropy analysis
  13. Non-contractible loops
  14. Entanglement entropy analysis for one non-contractible loop
  15. Configurations with Wilson lines
  16. ...and 16 more sections

Figures (28)

  • Figure 1: Cell decomposition of the disk. The partition can also be thought as the insertion of a Wilson line, with the corresponding group representation indices shown. The empty disk partition function is recovered when the representation of the Wilson line is considered to be trivial.
  • Figure 2: Cell decomposition of the cylinder. For later use we show the setup with one contractible Wilson loop.
  • Figure 3: A genus-1 cell may be built by gluing two pairs of pants as depicted in (\ref{['subfig:genus1cell']}). Each pair of pants may be cut open and represented as a heptagon. In (\ref{['subfig:plaquettepairofpants']}) we depict the plaquette obtained by cutting the left pair of pants along $X$ and $Y$.
  • Figure 4: Logarithm of the entropy as a function of genus for different gauge groups. The total area is fixed, $\varrho = 20$.
  • Figure 5: Entropy (\ref{['eq:1ConLoopEntropy']}) for varying area fraction for the loop in some $SU(2)$ representations calculated for intermediate (\ref{['subfig:cylCWL_fixed_tot_su2a']}) and large (\ref{['subfig:cylCWL_fixed_tot_su2b']}) total area $\varrho_t$. The vertical dashed lines correspond to the same fraction of the loop area and the total area on both plots.
  • ...and 23 more figures