Entanglement entropy of SU(3) Yang-Mills theory

TL;DR

The paper investigates the entanglement entropy of the SU(3) Yang–Mills vacuum using lattice Monte Carlo simulations and the replica-trick method. By computing free-energy differences via an interpolating action on multi-cut geometries, it extracts the l‑dependence of the entanglement entropy S_A and its derivative. At small l, ∂S_A/∂l scales as ~1/l^3 (implying S_A ~ 1/l^2), while ∂S_A/∂l vanishes around l* ≈ 0.6–0.7 fm, signaling a mass gap and a possible identification l* ~ 1/T_c. The entropic C-function is nonzero for l < l*, but large statistical errors above l* prevent a precise determination. Overall, the results support a confinement-related structure in entanglement, with future improvements anticipated from RG-improved actions.

Abstract

We calculate the entanglement entropy using a SU(3) quenched lattice gauge simulation. We find that the entanglement entropy scales as $1/l^2$ at small $l$ as in the conformal field theory. Here $l$ is the size of the system, whose degrees of freedom is left after the other part are traced out. The derivative of the entanglement entropy with respect to $l$ hits zero at about $l^{\ast} = 0.6 \sim 0.7$ [fm] and vanishes above the length. It may imply that the Yang-Mills theory has the mass gap of the order of $1/l^{\ast}$. Within our statistical errors, no discontinuous change can be seen in the entanglement entropy. We discuss also a subtle point appearing in gauge systems when we divide a system with cuts.

Figures (8)

• Figure 1: Schematic picture of the entanglement entropy predicted by the holographic approach for the AdS bubble solution. At small $l$, $\partial S_A/\partial l$ behaves as $1/l^3$ as conformal field theories in $(3+1)$-dimensional spacetime. By contrast, it vanishes at large $l$ where disconnected surfaces dominate.
• Figure 2: The complementary regions $A$ and $B$ separated by an imaginary boundary at $x=l$. $y$ and $z$ axes are perpendicular to the plane. Separation is purely an imaginary process and nothing has to be done on the physical state. The entanglement entropy measures quantum correlation between two regions $A$ and $B$.
• Figure 3: Schematic figure of spin system with a finite correlation length $\xi$. Spin degrees of freedom in two regions $A$ and $B$ separated more than the correlation length do not have quantum correlations, and do not contribute to quantum entanglement.
• Figure 4: Schematic picture for the system with two cuts in $x-t$ plane. In the region $A$ ($B$), the periodic boundary condition is imposed with the period $2/T$ ($1/T$).
• Figure 5: The difference $\langle S_{l=2} - S_{l=1} \rangle$ on $16^3 \times 32$ at $\beta=6.0$. The integration from $\alpha=0$ to $\alpha=1$ gives $\partial S_A/\partial l$ at $l=3a/2$, the midpoint between $l=a$ and $l=2a$.