Topological structure of the entanglement radius of Yang-Mills flux tubes

Abstract

We expand on recent work arXiv:2601.17199 demonstrating the existence of a novel entanglement radius $ξ_0$ characterizing flux tube entanglement entropy (FTE$^2$) in (2+1)D Yang-Mills theory. This physical scale corresponds to the intrinsic thickness of the flux tube that must be fully severed by an entangling region for color degrees of freedom in the flux tube to contribute non-zero FTE$^2$. We consider here geometries of the entanglement region $V$ on the lattice where the length of the region cross-cutting the flux tube is of the same magnitude as $ξ_0$. Our results further the conclusions of arXiv:2601.17199 by adding detailed new information on the topological structure of the entanglement radius of color flux tubes.

Figures (6)

• Figure 1: Correlator of Polyakov loops representing quark (red) and antiquark (dark blue) sources in a $q=2$ replica $L_s\times L_\tau=6\times3$ lattice with region $V$ (shaded blue) having length $L_x=2a$. The lattice has periodicity $L_s$ in the spatial direction. The lattice is $L_\tau$-periodic in region $\bar{V}$ and $2L_\tau$-periodic in region $V$. Dashed gauge links and open vertices are images, respectively, of solid gauge links and filled vertices, determined by the respective temporal or spatial periodic boundary conditions.
• Figure 2: (Left) A depiction of the "half-slab" geometry of Ref. Amorosso:2024leg, with region $V$ (shaded blue) having length $L_x$ equal to half of the $x$-extent of the lattice. (Right) A depiction of the "small-slab" geometry, with region $V$ (shaded blue) of variable length $L_x$ in the $x$ direction. The quark and antiquark sources are separated by distance $L$ and located at $(0,\pm L/2)$.
• Figure 3: Examples of different overlaps of the flux tube with the entangling region $V$ (shaded blue). (Left) The flux tube exhibits a full boundary crossing and contributes $2\ln 2$ to FTE$^2$ . (Center) The center of the slab $V$ is too far to the right ($x$ too large) to fully sever the flux tube; the flux tube exhibits a partial boundary crossing and does not contribute to FTE$^2$ . (Right) The length ($L_x$) of the slab $V$ is too small relative to $\xi_0$; the flux tube exhibits a partial boundary crossing and does not contribute to FTE$^2$ .
• Figure 4: Dependence of FTE$^2$$\tilde{S}_{\vert{Q\bar{Q}}}$ on the transverse position of the slab $x$ computed with small-slab geometry (Fig. \ref{['fig:refinedsmallslab']}), in linear (left) and logarithmic (right) scale. Calculations are done in (2+1)D $SU(2)$ Yang-Mills theory with ${Q\bar{Q}}$ distance fixed to $L\sqrt{\sigma_0}\approx0.67$. The solid line shows the fit to a Gaussian shape.
• Figure 5: (Left) Plot of FTE$^2$ as a function of $x$ with varying values of the quark separation $L$. Note that the FTE$^2$ value of the peak of each profile decreases as $L$ increases. (Right) FTE$^2$ as a function of $x$ with varying values of the slab length $L_x$. The FTE$^2$ value of the peak of each profile increases significantly as $L_x$ increases. At the other extreme $L_x\sqrt{\sigma_0}=0.22$, the FTE$^2$ contribution at $x=0$ is still non-zero despite $L_x < 2\xi_0\sim0.37\sigma_0^{-1/2}$.