Rényi exponent landscape of multipartite entanglement in free-fermion systems

We show that the Rényi tripartite information $I_3^{(\alpha )}$ of free fermions exhibits a qualitatively $\alpha$-dependent scaling at small Fermi momentum, in sharp contrast to bipartite entropy where only the prefactor changes. In the rank-1 regime ($z=k_{F}w\ll 1$), $I_3^{(\alpha )}$ receives contributions from two competing channels – a fractional-moment channel $\sim z^{\alpha }$ (active for non-integer $\alpha$) and a polynomial channel $\sim z^m$ from the first nonvanishing inclusion-exclusion moment ${\sigma }_m$ – yielding the scaling exponent ${\beta }_m(\alpha )=\min (\alpha ,m)$ for $m$-partite information of $m$ adjacent strips. Integer Rényi indices $\alpha =2,3,\dots$ are anomalous: the fractional channel closes and the exponent jumps to $m$ or higher. A direct consequence is a replica obstruction: $I_m^{(n)}\mathrm{/}{I}_m^{(1)}\sim z^{m-1}\to 0$ for all integer $n\ge 2$, so the leading von Neumann signal cannot be reconstructed from integer Rényi data at the level of leading scaling – a situation with no bipartite analog. Conversely, negativity-based measures ($\alpha =1\mathrm{/}2$) give a $20\times$ enhanced signal compared to von Neumann. We derive the underlying product formula for the coefficient $c(w_A,w_B,w_D)$, prove an $m$-partite generating function for the inclusion-exclusion moments, and verify all results numerically to high precision.