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Revisiting Entanglement Entropy of Lattice Gauge Theories

Ling-Yan Hung, Yidun Wan

TL;DR

This work reexamines the Casini et al. claim that entanglement entropy in gauge theories is ambiguous due to boundary choices. It shows that in the Kitaev model—the prototypical example with a gauge-like structure but without true bulk gauge redundancy—the topological entanglement entropy is robust and equals $\\gamma=\\ln|G|$, independent of whether one uses electric or magnetic centers; the naive magnetic-center calculation yields $S_E = L\\ln|G| - \\ln|G|$, and duality ensures boundary-choice independence in the thermodynamic limit. The authors then argue that the perceived ambiguity in true gauge theories arises from how boundary degrees of freedom are counted and which observables are permitted, framing entanglement in terms of measurable, gauge-invariant boundary data. They extend the analysis to non-Abelian theories by transforming edge states to a representation basis and showing that boundary representation probabilities determine the entanglement, with fusion constraints at vertices shaping the entropy. Overall, the paper clarifies what entanglement entropy in gauge theories measures, highlights the role of boundary observables, and provides a coherent framework—including a non-Abelian generalization—for interpreting topological entanglement in gauge systems.

Abstract

Casini et al raise the issue that the entanglement entropy in gauge theories is ambiguous because its definition depends on the choice of the boundary between two regions.; even a small change in the boundary could annihilate the otherwise finite topological entanglement entropy between two regions. In this article, we first show that the topological entanglement entropy in the Kitaev model which is not a true gauge theory, is free of ambiguity. Then, we give a physical interpretation, from the perspectives of what can be measured in an experiement, to the purported ambiguity of true gauge theories, where the topological entanglement arises as redundancy in counting the degrees of freedom along the boundary separating two regions. We generalize these discussions to non-Abelian gauge theories.

Revisiting Entanglement Entropy of Lattice Gauge Theories

TL;DR

This work reexamines the Casini et al. claim that entanglement entropy in gauge theories is ambiguous due to boundary choices. It shows that in the Kitaev model—the prototypical example with a gauge-like structure but without true bulk gauge redundancy—the topological entanglement entropy is robust and equals , independent of whether one uses electric or magnetic centers; the naive magnetic-center calculation yields , and duality ensures boundary-choice independence in the thermodynamic limit. The authors then argue that the perceived ambiguity in true gauge theories arises from how boundary degrees of freedom are counted and which observables are permitted, framing entanglement in terms of measurable, gauge-invariant boundary data. They extend the analysis to non-Abelian theories by transforming edge states to a representation basis and showing that boundary representation probabilities determine the entanglement, with fusion constraints at vertices shaping the entropy. Overall, the paper clarifies what entanglement entropy in gauge theories measures, highlights the role of boundary observables, and provides a coherent framework—including a non-Abelian generalization—for interpreting topological entanglement in gauge systems.

Abstract

Casini et al raise the issue that the entanglement entropy in gauge theories is ambiguous because its definition depends on the choice of the boundary between two regions.; even a small change in the boundary could annihilate the otherwise finite topological entanglement entropy between two regions. In this article, we first show that the topological entanglement entropy in the Kitaev model which is not a true gauge theory, is free of ambiguity. Then, we give a physical interpretation, from the perspectives of what can be measured in an experiement, to the purported ambiguity of true gauge theories, where the topological entanglement arises as redundancy in counting the degrees of freedom along the boundary separating two regions. We generalize these discussions to non-Abelian gauge theories.

Paper Structure

This paper contains 7 sections, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Entanglement entropy in the Kitaev model
  3. Electric vs magnetic center in the Kitaev model
  4. Naive entanglement entropy with the magnetic center
  5. Entanglement entropy in "true" gauge theories vs SPT
  6. Generalization to Non-Abelian theories
  7. Conclusion

Figures (3)

  • Figure 1: (Color online) Electric (rough) boundary on the left and magnetic (smooth) boundary on the right. The boundaries are in dashed lines, which are part of region $V$.
  • Figure 2: (Color online) An infinite lattice divided into two regions: $A$, inside the red (thicker line) loop, and $\bar{A}$, outside of the loop. the blue (dashed) lines comprising the rough boundary also belong to $A$. The wiggly lines simply represent the parts of the infinite lattice that are neglected.
  • Figure 3: