Numerical study of entanglement entropy in SU(2) lattice gauge theory

TL;DR

This study numerically investigates the entanglement entropy between a three-dimensional slab and its complement in four-dimensional SU(2) lattice gauge theory. Using a replica-path-integral framework on lattices with cuts, the authors separate $S[A]$ into a UV-divergent part $S_{UV}$ and a UV-finite part $S_f(l)$, providing evidence for a nonanalytic derivative at a critical thickness $l_c$ near $0.5\,\mathrm{fm}$, in qualitative agreement with holographic predictions of a connected/disconnected minimal-surface transition. At finite temperature, the nonanalytic behavior appears to persist and may turn into a crossover above $T_c$, suggesting a line in the $(l,T)$ plane. The results reinforce the holographic picture of confinement affecting the entanglement structure and demonstrate the utility of lattice entanglement measures as probes of colour confinement and the emergence of colourless degrees of freedom.

Abstract

The entropy of entanglement between a three-dimensional slab of thickness l and its complement is studied numerically for four-dimensional SU(2) lattice gauge theory. We find a signature of a nonanalytic behavior of the entanglement entropy, which was predicted recently for large N_c confining gauge theories in the framework of AdS/CFT correspondence. The derivative of the entanglement entropy over l is likely to have a discontinuity at some l = l_c. It is argued that such behavior persists even at finite temperatures, probably turning into a sort of crossover for temperatures larger than the temperature of the deconfinement phase transition. We also confirm that the entanglement entropy contains quadratically divergent l-independent term, and that the nondivergent terms behave as the inverse square of l at small distances.

Figures (9)

• Figure 1: The topology of space on which the free energy $F \left[ A, s, T \right]$ in (\ref{['ent_vs_fe']}) is calculated, an example for $s = 2$. Dashed lines with arrows denote identification of cut sides, i.e. periodic boundary conditions in time direction.
• Figure 2: The distributions of average action density in the $\left( x^{0},x^{1} \right)$ plane for different lattices and cut sizes: at the upper left: $24 \times 20^{3}$ lattice with two cuts, $a^{-1} l = 8$, $a = 0.12 \, fm$, at the upper right: $24 \times 20^{3}$ lattice with two cuts, $a^{-1} l = 6$, $a = 0.12 \, fm$, at the lower left: $28 \times 24^{3}$ lattice with two cuts, $a^{-1} l = 6$, $a = 0.10 \, fm$, at the lower right: $18 \times 12^{3}$ lattice with three cuts, $a^{-1} l = 6$, $a = 0.17 \, fm$. The cuts are shown as thick solid lines. The topology of lattices with two cuts is illustrated on Fig. \ref{['fig:cutted_box']}.
• Figure 3: Average excess of action on the lattice plaquettes which are closest to the branching points on different lattices with two and three cuts.
• Figure 4: The dependence of the derivative of the entanglement entropy $\frac{1}{|\partial A|} \, \frac{S_{f} \left( l \right) }{\partial l}$ on $l$. Solid line is the fit of the data by the function $C \, l^{-3}$.
• Figure 5: The discontinuity of the derivative of the entanglement entropy over $l$ near $l_{c} \approx 0.5 \, fm$.