Entanglement Enabled Tomography of Flux Tubes in (2+1)D Yang-Mills Theory

TL;DR

The paper analyzes the entanglement structure of color flux tubes between static quarks in (2+1)D SU($N_c$) Yang-Mills via Flux Tube Entanglement Entropy (FTE^2). Using a lattice replica method with Polyakov-loop observables, the authors separate vibrational and internal color contributions to FTE^2 and identify a novel topological scale, the entanglement radius $\xi_0$, which must be fully severed by the entangling region to generate color entanglement. The study finds that $\xi_0$ grows approximately linearly with the number of colors $N_c$ and is independent of the Rényi index $q$ and inter-quark separation $L$, while the tail scale $\lambda$ remains roughly constant and correlates with the inverse glueball mass. These results indicate a bulk, topological origin for the color entanglement in confining gauge theories, challenging simple one-dimensional string pictures and informing effective models of flux tubes and confinement.

Abstract

We investigate the entangling properties of the color flux tube between a static quark-antiquark pair in pure gauge Yang-Mills theory. In earlier works, we defined a gauge-invariant flux tube entanglement entropy (FTE$^2$), the excess entanglement entropy of a region of gluon fields that can be attributed to the color flux tube, and demonstrated that it is finite in the continuum limit. FTE$^2$ was shown to have two contributions, one from the vibrations of the QCD string, and the other from its internal (color) degrees of freedom. In this work, we further explore the internal color component in (2+1)D Yang-Mills theory for $SU(N_c)$ gauge groups, varying $2\le N_c\le5$. We identify a novel physical scale in the theory, the entanglement radius $ξ_0$. This radius characterizes the transverse extent of the flux tube that must be completely severed by an entangling region to capture the entanglement entropy of color degrees of freedom. The key feature underlying this phenomenon is its topological nature. This is revealed through systematic studies of multi-slab entangling regions in which FTE$^2$ changes sharply when boundaries of the slabs completely cross-cut the flux tube. We find that $ξ_0$ increases approximately linearly with $N_c$ and is independent of both Rényi replica number and the inter-quark separation length. We also study FTE$^2$ as a function of the entangling region's transverse displacement from the static quark pair and observe behavior consistent with a previously identified intrinsic width $λ$ of the flux tube, with an extracted value in agreement with the inverse mass of the lightest glueball for the gauge groups studied.

Figures (15)

• Figure 1: Lattice implementation of $\text{Tr}(\rho_V^q)$ for $q=2$ replicas of an $L_x\times L_t=6\times3$ lattice. The spatial boundary points have periodicity $L_x$. The temporal boundary conditions have periodicity $qL_t$ in region $V$ (shaded blue) and $L_t$ in $\bar{V}$. Open points and dashed links are images of the solid points and links determined by the spatial and temporal periodic boundary conditions. To calculate FTE$^2$, one must add temporal Polyakov loops to region $\bar{V}$.
• Figure 2: Flux tube entanglement entropy was computed in Ref. Amorosso:2024leg using the "half-slab" geometry depicted above. Region $V$ (shaded blue) has width $w$ in the $y$-direction and length $L_x/2$ in the $x$-direction. The flux tube is centered at $(0,0)$ with quark and antiquark separated by distance $L$ in the $y$-direction.
• Figure 3: (Left) FTE$^2$ of the half-slab geometry in $SU(2)$ (2+1)D Yang-Mills as a function of the lattice spacing $a$ and slab edge position $x$. Results are shown for a fixed inter-quark distance $L\sqrt{\sigma_0}=0.67$, slab width $w\sqrt{\sigma_0}=0.22$, and $y=0$. (Right) The offset of the half-maximum value of internal FTE$^2$ from $x=0$ (the value $x=-\xi_0$ at which $\tilde{S}=\ln2$) as a function of the lattice spacing. Note that $\xi_0$ converges to a finite value as we approach the continuum limit.
• Figure 4: (Left) Schematic of a flux tube exhibiting a full boundary crossing in the staggered two-slab geometry with $\Delta y=0$. Here the flux tube has two disconnected components in $\bar{V}$. (Center) Schematic of a flux tube exhibiting a partial boundary crossing in the staggered two-slab geometry with $\Delta y>0$. Here the flux tube has one connected component in $\bar{V}$. (Right) FTE$^2$ of the staggered two-slab geometry in $SU(2)$ Yang-Mills as a function of $x$ for different gap sizes $\Delta y$. A steep drop in FTE$^2$ is seen for non-zero $\Delta y$ at $x=0$. Here $a\sqrt{\sigma_0}=0.056$, the individual slab width is $4a$, and the quark separation $L=20a$. The proposed fit form (\ref{['eq:roughEst']}) (plus a fitted constant to account for vibrational contributions) is shown with a solid line.
• Figure 5: Schematic of two flux tubes with the same deflection (partially) intersecting a region $V$ (shaded blue). (Left) The flux tube has zero entanglement radius and exhibits a full boundary crossing, being separated into two disconnected components within $\bar{V}$ and resulting in an internal entropy of $2\ln N_c$. (Right) The flux tube has finite non-zero entanglement radius and exhibits a partial boundary crossing. The flux tube has one connected component within $\bar{V}$, resulting in zero internal entanglement entropy.