Tripartite information of free fermions: a universal entanglement coefficient from the sine kernel

TL;DR

This paper derives a universal scaling function for the tripartite information in free fermions, revealing a zero that distinguishes monogamy violation and satisfaction, with implications for entanglement and Lifshitz transitions.

Contribution

It provides an exact analytical form of the universal function g(z) for tripartite information in free fermions, connecting sine kernel spectra to entanglement properties.

Findings

g(z) has a zero at z* ≈ 1.329, separating monogamy violation and satisfaction.

The coefficient c in g(z) is approximately 0.2747, derived from sine kernel limits.

The linear coefficient in g(z) is independent of the Luttinger parameter K in interacting models.

Abstract

We study the tripartite information I_3 of free fermions on two-dimensional lattices partitioned into three adjacent strips of width w. Translation invariance yields the exact decomposition I_3 = \sum_{k_y} g(k_F(k_y) w), where g(z) is a universal function of the scaling variable z = k_F w, determined by the spectrum of the sine-kernel (Slepian) integral operator. We find that g(z) has a unique zero at z* = 1.329 +/- 0.001: modes with k_F w < z* violate monogamy of mutual information (g > 0), while modes with k_F w > z* satisfy it (g < 0). The central analytical result is g(z) = cz + O(z^3 ln z) with c = 3 ln(4/3)/pi ~ 0.2747, derived from the rank-1 limit of the sine kernel. Two exact cancellations -- of the z ln z area-law terms and of the z^2 terms -- are intrinsic to the I_3 combination. The coefficient c generalizes to n-partite information: c_n = (n/pi) ln R_n with R_n a rational number from binomial combinatorics. For Renyi entropy of index alpha, we show that g_alpha(z) ~ z^alpha for alpha < 2 and g_2(z) = -(8/pi^3)z^3: von Neumann entropy (alpha = 1) uniquely gives linear sensitivity to Lifshitz transitions, while Renyi-2 gives only cubic sensitivity. We verify all predictions on square, triangular, and cubic lattices. DMRG calculations on the interacting t-V model provide evidence that the linear coefficient is independent of the Luttinger parameter K.