On The Entanglement Entropy For Gauge Theories

TL;DR

This work provides a rigorous, lattice-based definition of entanglement entropy for gauge theories by embedding the physical, gauge-invariant state into a larger tensor-product Hilbert space and tracing over the outside links. The resulting entropy, $S_{EE}$, is shown to reproduce the electric-centre structure of Casini–Huerta–Rosabal for Abelian cases and to align with replica-trick calculations in both Abelian and Non-Abelian settings, while not coinciding with Bell-pair distillation paradigms due to superselection sectors. The authors illustrate the construction through $\mathbb{Z}_2$, $U(1)$, and $SU(2)$ lattice gauge theories, discuss operational measurability of $\rho_{in}$, and compare with the extended lattice construction (ELC), clarifying when different definitions agree or differ. The results provide a robust framework for entanglement in gauge theories and suggest avenues for exploring spatial-region entanglement, continuum limits, and holographic connections via replica-trick logic.

Abstract

We propose a definition for the entanglement entropy of a gauge theory on a spatial lattice. Our definition applies to any subset of links in the lattice, and is valid for both Abelian and Non-Abelian gauge theories. For $\mathbb{Z}_N$ and $U(1)$ theories, without matter, our definition agrees with a particular case of the definition given by Casini, Huerta and Rosabal. We also argue that in general, both for Abelian and Non-Abelian theories, our definition agrees with the entanglement entropy calculated using a definition of the replica trick. Our definition, however, does not agree with some standard ways to measure entanglement, like the number of Bell pairs which can be produced by entanglement distillation.

Figures (4)

• Figure 1: The subset of links represented by solid lines constitute the inside. The remaining links shown by dashed lines are the outside.
• Figure 2: The lattice with boundary vertex $V_1$. Links 1 and 2 are in the outside subset, while links 3 and 4 are in the inside subset.
• Figure 3: The red curve is the boundary of the spatial region of interest. Several links intersect it, including link A which emanates from vertex $V_A$ and and links F,G, which emanate from vertex $V_E$.
• Figure 4: The extended lattice construction. Blue dots represent additional vertices which have been introduced at the points where links of the original lattice intersect the boundary shown in red. Links touching the blue dots are the new links which have been introduced.