Tripartite information of two-dimensional free fermions: a sine-kernel spectral constant from Fermi surface geometry

Abstract

We show that monogamy of mutual information (MMI) in free-fermion ground states is a property of the observation scale, not of the quantum state. For three adjacent strips of width $w$ on a two-dimensional lattice, translation invariance decomposes the tripartite information as $I_3 = \sum_{k_y} g(k_F(k_y)\, w)$, where $g(z)$ is a universal function of the dimensionless product $z = k_F w$, determined by the spectrum of the sine-kernel integral operator (the Slepian concentration operator). We prove that $g(z)$ has a unique zero at $z^* \approx 1.329$: modes with $k_F w < z^*$ violate MMI ($g > 0$), while modes with $k_F w > z^*$ satisfy it ($g < 0$). Since $z^* / k_F w \to 0$ as $w \to \infty$, any Fermi surface eventually satisfies MMI at large $w$, while any gapless system violates it at sufficiently small $w$. The classification of states as "holographic" or "non-holographic" by the sign of $I_3$ is thus scale-dependent. We establish the properties of $g(z)$ analytically and show that $z^*$ is determined to $0.12\%$ by the cancellation of only two Slepian eigenvalue contributions. For Rényi entropies with index $α> 1$, the function $g_α(z)$ oscillates with multiple sign changes. We verify the framework on square and triangular lattices and show that interactions shift $z^*$ by $\sim 1$--$2\%$.

Figures (6)

• Figure 1: Fermi surfaces at half filling for: (a) square lattice, $t' = 0$ (perfect nesting); (b) $t' = 0.12 \approx t'_*$; (c) $t' = 0.30$ (strong nesting breaking); (d) triangular lattice (no nesting). The value of $I_3/L$ (at $L=256$, $w=2$) is indicated below each panel.
• Figure 2: Phase diagram of $I_3/L$ in the $(w, t')$ plane for the square lattice at half filling ($L = 128$). Red: $I_3 > 0$ (MMI violated). Blue: $I_3 < 0$ (MMI satisfied). The solid black curve marks the $I_3 = 0$ boundary. The MMI-violating region is confined to small $t'$ and small $w$, disappearing entirely for $t' \gtrsim 0.08$; the range extends to $t' = 0.5$ (van Hove singularity), confirming that no reentrant MMI violation occurs.
• Figure 3: (a) $I_3$ per $k_y$ mode for the square lattice ($L = 128$, $w = 2$) at $t' = 0$ (blue) and $t' = 0.20$ (red). (b) Effective filling fraction $\nu(k_y)$ for the same parameters. Green shading marks the region $\nu < 0.19$ or $\nu > 0.81$ where $I_3^{(1D)}(\nu) > 0$ (see Fig. \ref{['fig:universal_curve']}).
• Figure 4: Universal 1D tripartite information $I_3^{(1D)}(\nu)$ as a function of filling fraction for three strip widths ($L = 256$). The curve is symmetric about $\nu = 1/2$ by particle-hole symmetry. Dashed lines mark $\nu_* \approx 0.19$ for $w = 2$. As $w$ increases, $\nu_*$ decreases and the positive region shrinks.
• Figure 5: (a) Analytical $I_3^{(1D)}(\nu)$ from Eq. (\ref{['eq:I3_exact']}) for strip widths $w = 1$--$16$. (b) The product $k_F^* \times w$ converges to $z^* \approx 1.329$ (dashed red); the value $4/3$ is shown for comparison (dotted blue). (c) Scaling collapse: $I_3$ plotted against $z = k_F w$ for various $w$, confirming convergence to a universal curve.