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Entanglement Entropy and Duality

Djordje Radicevic

TL;DR

The paper investigates entanglement entropy through an algebraic lens, demonstrating that entropies remain invariant across dualities when subsystems are defined by non-maximal algebras and when one accounts for edge sectors or winding configurations. By analyzing Ising-Ising, Ising-gauge, and bosonization dualities on lattices, it shows that maximal algebras map to non-maximal ones under duality, and that tracing out external degrees of freedom is equivalent to tracing the full density operator with a sum over superselection sectors. The results reveal that entropy can originate from either edge modes or global degeneracies, depending on the dual frame, highlighting a UV/IR correspondence intrinsic to dualities. These insights pave the way for applying the algebraic entropy framework to more intricate dualities, including holography, where bulk-boundary entanglement matching may require careful treatment of edge and sector contributions.

Abstract

Using the algebraic approach to entanglement entropy, we study several dual pairs of lattice theories and show how the entropy is completely preserved across each duality. Our main result is that a maximal algebra of observables in a region typically dualizes to a non-maximal algebra in a dual region. In particular, we show how the usual notion of tracing out external degrees of freedom dualizes to a tracing out coupled to an additional summation over superselection sectors. We briefly comment on possible extensions of our results to more intricate dualities, including holographic ones.

Entanglement Entropy and Duality

TL;DR

The paper investigates entanglement entropy through an algebraic lens, demonstrating that entropies remain invariant across dualities when subsystems are defined by non-maximal algebras and when one accounts for edge sectors or winding configurations. By analyzing Ising-Ising, Ising-gauge, and bosonization dualities on lattices, it shows that maximal algebras map to non-maximal ones under duality, and that tracing out external degrees of freedom is equivalent to tracing the full density operator with a sum over superselection sectors. The results reveal that entropy can originate from either edge modes or global degeneracies, depending on the dual frame, highlighting a UV/IR correspondence intrinsic to dualities. These insights pave the way for applying the algebraic entropy framework to more intricate dualities, including holography, where bulk-boundary entanglement matching may require careful treatment of edge and sector contributions.

Abstract

Using the algebraic approach to entanglement entropy, we study several dual pairs of lattice theories and show how the entropy is completely preserved across each duality. Our main result is that a maximal algebra of observables in a region typically dualizes to a non-maximal algebra in a dual region. In particular, we show how the usual notion of tracing out external degrees of freedom dualizes to a tracing out coupled to an additional summation over superselection sectors. We briefly comment on possible extensions of our results to more intricate dualities, including holographic ones.

Paper Structure

This paper contains 9 sections, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Entropy in a single spin
  3. Ising-Ising duality ($d = 1$)
  4. Ising-gauge duality ($d = 2$)
  5. Setup
  6. Entanglement in gauge theory
  7. Entanglement in the dual picture
  8. Bosonization ($d = 1$)
  9. Outlook

Figures (2)

  • Figure 1: (color online) KW duality in $d = 1$. Above: the original picture. Thick red dots are the set $V$, and operators generating its maximal algebra are explicitly labeled. Below: dual picture. Black circles denote edge sites without $\tau^z$ operators; all operators in the dual algebra are also labeled.
  • Figure 2: (color online) Duality in $d = 2$: Thick black lines are the set $V$, and grey circles denote edge sites in $\partial V$. Electric operators $\sigma^x$ are defined on all thick black lines, and magnetic operators $W$ are defined on all thick black plaquettes. All red circles together form the dual set $\widetilde{V}$. Operators $\tau^z$ on filled-in circles belong to the dual algebra $\widetilde{\mathcal{A}}_{\widetilde{V}}$, as do all $\tau^x\tau^x$ pairs on sites connected by red lines.