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Rényi exponent landscape of multipartite entanglement in free-fermion systems

Aleksandrs Sokolovs

Abstract

We show that the Rényi tripartite information $I_3^{(α)}$ of free fermions exhibits a qualitatively $α$-dependent scaling at small Fermi momentum, in sharp contrast to bipartite entropy where only the prefactor changes. In the rank-1 regime ($z = k_F w \ll 1$), $I_3^{(α)}$ receives contributions from two competing channels -- a fractional-moment channel $\sim z^α$ (active for non-integer $α$) and a polynomial channel $\sim z^m$ from the first nonvanishing inclusion-exclusion moment $σ_m$ -- yielding the scaling exponent $β_m(α) = \min(α, m)$ for $m$-partite information of $m$ adjacent strips. Integer Rényi indices $α= 2, 3, \ldots$ 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 Rényi data at the level of leading scaling -- a situation with no bipartite analog. Conversely, negativity-based measures ($α= 1/2$) 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.

Rényi exponent landscape of multipartite entanglement in free-fermion systems

Abstract

We show that the Rényi tripartite information of free fermions exhibits a qualitatively -dependent scaling at small Fermi momentum, in sharp contrast to bipartite entropy where only the prefactor changes. In the rank-1 regime (), receives contributions from two competing channels -- a fractional-moment channel (active for non-integer ) and a polynomial channel from the first nonvanishing inclusion-exclusion moment -- yielding the scaling exponent for -partite information of adjacent strips. Integer Rényi indices are anomalous: the fractional channel closes and the exponent jumps to or higher. A direct consequence is a replica obstruction: for all integer , 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 () give a enhanced signal compared to von Neumann. We derive the underlying product formula for the coefficient , prove an -partite generating function for the inclusion-exclusion moments, and verify all results numerically to high precision.
Paper Structure (11 sections, 1 theorem, 6 equations, 1 figure, 1 table)

This paper contains 11 sections, 1 theorem, 6 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Setup
  3. Master asymptotic formula
  4. Rényi exponent landscape
  5. Consequences
  6. Replica obstruction
  7. Negativity enhancement
  8. Rényi-2 blindness grows with $m$
  9. Product formula for the von Neumann coefficient
  10. Discussion
  11. Conclusions

Key Result

Theorem 1

For $m$-partite information of $m$ adjacent strips: At integer $\alpha = n \geq 2$, the fractional channel closes and $\beta_m(n) \geq m$, with anomalies: $\beta_3(2) = 3$, $\beta_3(3) = 4$; $\beta_4(2) = 4$, $\beta_4(3) = 4$, $\beta_4(4) = 5$.

Figures (1)

  • Figure 1: (a) $\beta(\alpha)$ for $I_3$ ($m = 3$). Dots: numerical. Line: $\min(\alpha, 3)$. Red circles: integer anomalies at $\alpha = 2$ ($\beta = 3$) and $\alpha = 3$ ($\beta = 4$). (b) $m$-partite generalization: $\beta_m(\alpha) = \min(\alpha, m)$, crossover at $\alpha = m$.

Theorems & Definitions (2)

  • Theorem 1: Exponent landscape
  • proof