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Parity superselection obstructs monogamy of mutual information in free fermions

Aleksandrs Sokolovs

Abstract

We prove that free fermions in the spin (tensor product) factorization violate monogamy of mutual information: $I_3^{\mathrm{spin}} > 0$ for three adjacent strips of width $w = 1, 2$ at all Fermi momenta, and for all~$w$ at $z = k_F w < z^* \approx 1.329$. The proof rests on an exact operator identity -- the fermionic and spin reduced density matrices of disjoint regions differ by the parity insertion $(-1)^{N_B}$ in the partial trace -- and a rigorous entropy bound. DMRG calculations on the $t$-$V$ chain quantify the effect for interacting fermions: the factorization contribution to the apparent $K$-dependence of $I_3$ exceeds the genuine interaction contribution by a factor of~8 at moderate filling, and accounts for ${\sim}80\%$ of the deviation observed in spin-basis numerics. Strong repulsion ($K \lesssim 0.7$) restores monogamy in both algebras. These results imply that any use of $I_3$ as a diagnostic -- whether for holographic duality, quantum chaos, or Fermi surface topology -- must specify the operator algebra; without this specification, the sign of $I_3$ is ambiguous.

Parity superselection obstructs monogamy of mutual information in free fermions

Abstract

We prove that free fermions in the spin (tensor product) factorization violate monogamy of mutual information: for three adjacent strips of width at all Fermi momenta, and for all~ at . The proof rests on an exact operator identity -- the fermionic and spin reduced density matrices of disjoint regions differ by the parity insertion in the partial trace -- and a rigorous entropy bound. DMRG calculations on the - chain quantify the effect for interacting fermions: the factorization contribution to the apparent -dependence of exceeds the genuine interaction contribution by a factor of~8 at moderate filling, and accounts for of the deviation observed in spin-basis numerics. Strong repulsion () restores monogamy in both algebras. These results imply that any use of as a diagnostic -- whether for holographic duality, quantum chaos, or Fermi surface topology -- must specify the operator algebra; without this specification, the sign of is ambiguous.
Paper Structure (11 sections, 2 theorems, 16 equations, 1 figure, 3 tables)

This paper contains 11 sections, 2 theorems, 16 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Setup
  3. The superselection defect identity
  4. Entropy bound
  5. Proof of positivity
  6. Analytical proof for $w = 1$
  7. Certified computation for $w = 2$
  8. Numerical verification for $w = 3$
  9. Physical picture
  10. Discussion
  11. Conclusion

Key Result

Proposition 1

For any state on adjacent blocks $A$, $B$, $D$: Equivalently, as a single operator equation: where $\Gamma_D = (-1)^{N_D}$ is the $D$-parity grading operator.

Figures (1)

  • Figure 1: (a) The fermionic $I_3^{\mathrm{ferm}} = g(z)$ (blue) changes sign at $z^* \approx 1.329$, while $I_3^{\mathrm{spin}}$ (green) remains positive for all $z$. The difference $\Delta S_{AD}$ (orange dashed) grows monotonically and overwhelms $|g|$ for $z > z^*$. (b) For $z > z^*$: the Gaussian budget $\mathcal{B}(z)$ (Theorem 1, purple) exceeds $|g(z)|$ (red shaded) with minimum margin $2.9\times$. Data for $w = 2$ (representative; $w = 1, 3$ show the same qualitative behavior with margins in Table 1).

Theorems & Definitions (4)

  • Proposition 1: Superselection defect identity
  • Theorem 1: Gaussian budget bound
  • proof
  • Conjecture 1: Spectral inequality