Where Multipartite Entanglement Localizes: The Junction Law for Genuine Multi-Entropy

Abstract

We uncover a "junction law" for genuine multipartite entanglement, suggesting that in gapped local systems multipartite entanglement is controlled and effectively localized near junctions where subsystem boundaries meet. Using the Rényi-2 genuine multi-entropy $\mathrm{GM}^{(\mathtt{q})}_2$ as a diagnostic of genuine $\mathtt{q}$-partite entanglement, we establish this behavior in $(2+1)$-dimensional gapped free-fermion lattices with correlation length $ξ$. For partitions with a single junction, $\mathrm{GM}^{(\mathtt{q})}_2$ exhibits a universal scaling crossover in $L/ξ$, growing for $L\llξ$ and saturating to a $ξ$-dependent constant for $L\ggξ$, up to $\mathcal{O}(e^{-L/ξ})$ corrections. In sharp contrast, for partitions without a junction, $\mathrm{GM}^{(\mathtt{q})}_2$ is exponentially suppressed in $L/ξ$ and drops below numerical resolution once $L\ggξ$. We observe the same pattern for $\mathtt{q}=3$ (tripartite) and $\mathtt{q}=4$ (quadripartite) cases, and further corroborate this localization by translating the junction at fixed system size. We also provide a geometric explanation of the junction law in holography. Altogether, these results show that in this gapped free-fermion setting genuine multipartite entanglement is localized within a correlation-length neighborhood of junctions.

Figures (15)

• Figure 1: Junction vs. no-junction geometries. (a) Junction: subsystem boundaries meet at a single point; the shaded disk indicates the correlation-length scale $\xi$. (b) No-junction: four-strip partition $A,B,C,D$ with no point adjacent to all subsystems.
• Figure 2: Junction present: tripartite case ($\mathtt{q}=3$). Scaling collapse of the normalized genuine multi-entropy $\mathrm{GM}^{(3)}_2/\mathrm{GM}^{(3)}_{2,\infty}$ versus $L/\xi$ for single-junction partitions at several masses $m$. Top: half-filled ground states. Bottom: representative low-lying excited states. While excited states exhibit nonuniversal deviations in the crossover regime ($L/\xi \lesssim 1$), all data approach the same universal saturation for $L/\xi \gg 1$. A small number of isolated points (typically $\le 5$ per series) show finite-size deviations but do not affect the scaling collapse or the large-$L/\xi$ behavior. Full-scale plots including all data points are provided in Appendix G.
• Figure 3: Junction present: quadripartite case ($\mathtt{q}=4$). Scaling collapse of the two independent Rényi-2 four-partite diagnostics for single-junction quadripartitions in the ground state. Top:$\mathrm{GM}^{(4)}_2|_{a=1/3}/\mathrm{GM}^{(4)}_{2,\infty}|_{a=1/3}$. Bottom:$I_{3,n=2}/I_{3,n=2,\infty}$. In both panels, data for different $m$ collapse versus $L/\xi$ and saturate at large $L/\xi$. The bottom panel exhibits strong short-$L/\xi$ nonuniversal deviations, including negative values.
• Figure 4: No junction: genuine multipartite entanglement is exponentially suppressed. Rényi-2 quadripartite diagnostics $\mathrm{GM}^{(4)}_2|_{a=1/3}$ and $I_{3,n=2}$ for the junction-free partition in Fig. \ref{['fig:setup']}(b). Both decay rapidly with $L/\xi$ and are strongly suppressed for $L\gg \xi$.
• Figure 5: Moving junction configuration.$\mathrm{GM}^{(3)}_2$ of the half-filled ground state versus the junction position, i.e., the shift of the internal $B|C$ cut in Fig. \ref{['fig:setup']}(a), at fixed $L=40$. Colors label $m$. A plateau appears when the junction is farther than $\sim \xi \,(\simeq 2/m)$ from the boundaries, and $\mathrm{GM}^{(3)}_2$ is suppressed within $\sim\xi$ of a boundary.