R\'enyi exponent landscape of multipartite entanglement in free-fermion systems

Abstract

We show that the R\'enyi 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 R\'enyi indices $\alpha = 2, 3, \ldots$ are anomalous: the fractional channel closes and the exponent jumps to $m$ or higher. A direct consequence is a replica obstruction: $I_m^{(n)}/I_m^{(1)} \sim z^{m-1} \to 0$ for all integer $n \geq 2$, so the leading von Neumann signal cannot be reconstructed from integer R\'enyi data at the level of leading scaling -- a situation with no bipartite analog. Conversely, negativity-based measures ($\alpha = 1/2$) 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.