Entanglement in Weakly Coupled Lattice Gauge Theories

TL;DR

The paper directly computes entanglement entropy in lattice Yang–Mills theories and identifies a universal logarithmic term in the weak-coupling regime that scales with the gauge-group dimension as ΔS = (1/2) dim(G) log(e^2 r) in d=2. This term arises from the entanglement of the softest gauge-mode with the environment and matches results obtained via Maxwell-scalar duality in 2D, strengthening the view that entanglement entropy captures confinement physics. The analysis contrasts strong-coupling vanishing entropy with weak-coupling N^2 scaling in the planar limit, and provides a framework that connects edge modes, superselection sectors, and dual descriptions. The work sets the stage for continuum formulations, extensions to gauge-matter theories, and potential implications for gravity and topological entanglement phenomena.

Abstract

We present a direct lattice gauge theory computation that, without using dualities, demonstrates that the entanglement entropy of Yang-Mills theories with arbitrary gauge group $G$ contains a generic logarithmic term at sufficiently weak coupling $e$. In two spatial dimensions, for a region of linear size $r$, this term equals $\frac{1}{2} \dim(G) \log\left(e^2 r\right)$ and it dominates the universal part of the entanglement entropy. Such logarithmic terms arise from the entanglement of the softest mode in the entangling region with the environment. For Maxwell theory in two spatial dimensions, our results agree with those obtained by dualizing to a compact scalar with spontaneous symmetry breaking.

Figures (2)

• Figure 1: (color online, adapted from Radicevic:2014kqa) Three examples of boundary electric operators defined in eq. \ref{['bdry el ops']}. Full lines denote links in $V$ and dotted lines denote links in $\bar{V}$. Gray dots denote elements of $\partial V$. The boundary electric Casimirs $\mathbf{E}^2_i$, defined on each boundary site, depend on electric generators on all links that enter that site and belong to $V$ (thick red lines), and conversely for $\bar{\mathbf{E}}^2_i$ and the wavy blue lines. Instead of specifying values of electric boundary operators, superselection sectors can also be specified by values of magnetic boundary operators. One such operator is the Wilson loop along the green plaquette at the edge of $V$.
• Figure 2: An impressionistic depiction of the phase structure of $\mathbb Z_\kappa$ gauge theory in $d = 2$. The level $\kappa$ is an integer and is represented as continuous just for convenience. The thick line roughly connects critical couplings at which deconfinement happens (the transition is second order for $\kappa = 2$ but may be first order for other levels). The dashed line follows $\kappa = 1/g$, roughly indicating where the weakly coupled ground state crosses over from a noncompact photon to the topological state. The shaded region, given by $\kappa \gg 1$ and $g \gg 1/\kappa$, is where the theory looks like a compact Maxwell lattice theory. At $1 \gg g \gg 1/\kappa$ (the leftmost part of the shaded region) the theory confines at very large distances due to the Polyakov mechanism, and at distances below this confinement scale the theory behaves as noncompact Maxwell, just as in the Coulomb phase.