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Entanglement entropy in d+1 SU(N) gauge theory

Alexander Velytsky

TL;DR

This work analyzes entanglement entropy for subregions in SU(N) lattice gauge theories across dimensions using the replica trick and Migdal-Kadanoff approximations. It demonstrates a nonanalytic change in the RG flow of character-expansion coefficients governing the partition function in d>2, signaling a transition in the entanglement structure within the confinement phase and connecting the critical length scale to the finite-temperature transition scale. The exactly solvable d=1 case provides baseline behavior, while MK-based results in higher dimensions reveal a dimension- and N_c-dependent transition location, with caution about MK's limitations. The study suggests entanglement entropy as a diagnostic for confinement-related phase structure and encourages numerical verification, e.g., Monte Carlo simulations, and links to vortex-free-energy order parameters.

Abstract

We consider the entanglement entropy for a sub-system in d+1 dimensional SU(N) lattice gauge theory. The 1+1 gauge theory is treated exactly and shows trivial behavior. Gauge theories in higher dimensions are treated within Migdal-Kadanoff approximation. We consider the gauge theory in the confinement phase. We demonstrate the existence of a non-analytical change from the short distance to long distance form in the entanglement entropy in such systems (d>2) reminiscent of a phase transition. The transition is manifested in nontrivial change in the RG flow of the character expansion coefficients defining the partition function.

Entanglement entropy in d+1 SU(N) gauge theory

TL;DR

This work analyzes entanglement entropy for subregions in SU(N) lattice gauge theories across dimensions using the replica trick and Migdal-Kadanoff approximations. It demonstrates a nonanalytic change in the RG flow of character-expansion coefficients governing the partition function in d>2, signaling a transition in the entanglement structure within the confinement phase and connecting the critical length scale to the finite-temperature transition scale. The exactly solvable d=1 case provides baseline behavior, while MK-based results in higher dimensions reveal a dimension- and N_c-dependent transition location, with caution about MK's limitations. The study suggests entanglement entropy as a diagnostic for confinement-related phase structure and encourages numerical verification, e.g., Monte Carlo simulations, and links to vortex-free-energy order parameters.

Abstract

We consider the entanglement entropy for a sub-system in d+1 dimensional SU(N) lattice gauge theory. The 1+1 gauge theory is treated exactly and shows trivial behavior. Gauge theories in higher dimensions are treated within Migdal-Kadanoff approximation. We consider the gauge theory in the confinement phase. We demonstrate the existence of a non-analytical change from the short distance to long distance form in the entanglement entropy in such systems (d>2) reminiscent of a phase transition. The transition is manifested in nontrivial change in the RG flow of the character expansion coefficients defining the partition function.

Paper Structure

This paper contains 6 sections, 47 equations, 5 figures.

Table of Contents

  1. Introduction
  2. $SU(N)$ gauge theory in $d+1$ dimensions
  3. $d=1$ gauge theory
  4. $d\ge2$ gauge theory
  5. Analyzing the RG flow
  6. Discussion of the results

Figures (5)

  • Figure 1: $Z_n$ for $1+1$ dimensional gauge theory.
  • Figure 2: $Z_n$ for $2+1$ dimensional theory.
  • Figure 3: Migdal decimation flow for $3+1$ dimensional $SU(2)$ gauge theory. Projection to $c^s_{1/2}$ and $c^s_1$; $(\beta,\lambda)$ are indicated.
  • Figure 4: Migdal decimation flow for $2+1$ dimensional $SU(2)$ gauge theory. Projection to $c^s_{1/2}$ and $c^s_1$; $\lambda=1.1$, $\beta=3.0,3.1,3.2$ and $4.0$ (inlet).
  • Figure 5: $2+1$ dimensional symmetric box.