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arXiv:2603.03103

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

Aleksandrs Sokolovs

cond-mat.stat-mechquant-phSubmitted Mar 3, 2026• Announced Mar 4, 2026• 0 citations• Revised Mar 8, 2026 (v3)
v1 · Mar 3, 2026v2 · Mar 4, 2026v3 · Mar 8, 2026
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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.

Comments
12 pages, 7 tables, 4 figures. v3: added DMRG evidence for interaction-independence of c, expanded numerical protocol, softened claims per referee feedback. Ancillary files: Python code reproducing all results

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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.
We show that the Rényi tripartite information I3(α)I_3^{(α)}I3(α)​ of free fermions exhibits a qualitatively ααα-dependent scaling at small Fermi momentum, in sharp contrast to bipartite entropy where only the prefactor changes. In the rank-1 regime (z=kFw≪1z = k_F w \ll 1z=kF​w≪1), I3(α)I_3^{(α)}I3(α)​ receives contributions from two competing channels -- a fractional-moment channel ∼zα\sim z^α∼zα (active for non-integer ααα) and a polynomial channel ∼zm\sim z^m∼zm from the first nonvanishing inclusion-exclusion moment σmσ_mσm​ -- yielding the scaling exponent βm(α)=min⁡(α,m)β_m(α) = \min(α, m)βm​(α)=min(α,m) for mmm-partite information of mmm adjacent strips. Integer Rényi indices α=2,3,…α= 2, 3, \ldotsα=2,3,… are anomalous: the fractional channel closes and the exponent jumps to mmm or higher. A direct consequence is a replica obstruction: Im(n)/Im(1)∼zm−1→0I_m^{(n)}/I_m^{(1)} \sim z^{m-1} \to 0Im(n)​/Im(1)​∼zm−1→0 for all integer n≥2n \geq 2n≥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=1/2α= 1/2α=1/2) give a 20×20\times20× enhanced signal compared to von Neumann. We derive the underlying product formula for the coefficient c(wA,wB,wD)c(w_A, w_B, w_D)c(wA​,wB​,wD​), prove an mmm-partite generating function for the inclusion-exclusion moments, and verify all results numerically to high precision.
We consider macroscopically large 3-partitions (A,B,C)(A,B,C)(A,B,C) of connected subsystems $A\ensuremath{\cup}B\ensuremath{\cup}C$ in infinite quantum spin chains and study the R\'enyi-$\ensuremath{\alpha}$ tripartite information ${I}_{3}^{(\ensuremath{\alpha})}(A,B,C)$. At equilibrium in clean 1D systems with local Hamiltonians it generally vanishes. A notable exception is the ground state of conformal critical systems, in which ${I}_{3}^{(\ensuremath{\alpha})}(A,B,C)$ is known to be a universal function of the cross ratio x=∣A∣∣C∣/[(∣A∣+∣B∣)(∣C∣+∣B∣)]x=|A||C|/[(|A|+|B|)(|C|+|B|)]x=∣A∣∣C∣/[(∣A∣+∣B∣)(∣C∣+∣B∣)], where ∣A∣|A|∣A∣ denotes AAA's length. We identify different classes of states that, under time evolution with translationally invariant Hamiltonians, locally relax to states with a nonzero (R\'enyi) tripartite information, which furthermore exhibits a universal dependency on xxx. We report a numerical study of ${I}_{3}^{(\ensuremath{\alpha})}$ in systems that are dual to free fermions, propose a field-theory description, and work out their asymptotic behavior for $\ensuremath{\alpha}=2$ in general and for generic $\ensuremath{\alpha}$ in a subclass of systems. This allows us to infer the value of ${I}_{3}^{(\ensuremath{\alpha})}$ in the scaling limit $x\ensuremath{\rightarrow}{1}^{\ensuremath{-}}$, which we call ``residual tripartite information''. If nonzero, our analysis points to a universal residual value $\ensuremath{-}log2$ independently of the R\'enyi index $\ensuremath{\alpha}$, and hence applies also to the genuine (von Neumann) tripartite information.
Abstract We consider the Rényi-α tripartite information I3(α) {I}_3^{\left(\alpha \right)} I3(α)​ I 3 α of three adjacent subsystems in the stationary state emerging after global quenches in noninteracting spin chains from both homogeneous and bipartite states. We identify settings in which I3(α) {I}_3^{\left(\alpha \right)} I3(α)​ I 3 α remains nonzero also in the limit of infinite lengths and develop an effective quantum field theory description of free fermionic fields on a ladder. We map the calculation into a Riemann-Hilbert problem with a piecewise constant matrix for a doubly connected domain. We find an explicit solution for α = 2 and an implicit one for α > 2. In the latter case, we develop a rapidly convergent perturbation theory that we use to derive analytic formulae approximating I3(α) {I}_3^{\left(\alpha \right)} I3(α)​ I 3 α with outstanding accuracy.
In the presence of a sharp corner in the boundary of the entanglement region, the entanglement entropy (EE) and Rényi entropies for 3d CFTs have a logarithmic term whose coefficient, the corner function, is scheme-independent. In the limit where the corner becomes smooth, the corner function vanishes quadratically with coefficient σ for the EE and σn for the Rényi entropies. For a free real scalar and a free Dirac fermion, we evaluate analytically the integral expressions of Casini, Huerta, and Leitao to derive exact results for σ and σn for all n=2,3,… . The results for σ agree with a recent universality conjecture of Bueno, Myers, and Witczak-Krempa that σ/CT=π2/24 in all 3d CFTs, where CT is the central charge. For the Rényi entropies, the ratios σn/CT do not indicate similar universality. However, in the limit n→∞, the asymptotic values satisfy a simple relationship and equal 1/(4π2) times the asymptotic values of the free energy of free scalars/fermions on the n-covered 3-sphere.
We investigate multipartite information and entanglement measures in the ground state of a free-fermion model defined on a Hamming graph. Using the known diagonalization of the adjacency matrix, we solve the model and construct the ground-state correlation matrix. Moreover, we find all the eigenvalues of the chopped correlation matrix when the subsystem consists of n disjoint Hamming subgraphs embedded in a larger one. These results allow us to find an exact formula for the entanglement entropy of disjoint graphs, as well as for the mutual and tripartite information. We use the exact formulas for these measures to extract their asymptotic behavior in two distinct thermodynamic limits, and find excellent match with the numerical calculations. In particular, we find that the entanglement entropy admits a logarithmic violation of the area law which decreases the amount of entanglement compared to the area law scaling.
We prove that free fermions in the spin (tensor product) factorization violate monogamy of mutual information: I3spin>0I_3^{\mathrm{spin}} > 0I3spin​>0 for adjacent strips of width w≤6w \le 6w≤6 at all Fermi momenta, and for all www at z=kFw<z∗≈1.329z = k_F w < z^* \approx 1.329z=kF​w<z∗≈1.329. Many-body computation at w=6w=6w=6 via the GGG-matrix formula maps the scaling-limit function I3spin(z)I_3^{\mathrm{spin}}(z)I3spin​(z): it has a minimum of 0.1000.1000.100 at z≈1.5z \approx 1.5z≈1.5, numerically establishing the conjecture I3spin>0I_3^{\mathrm{spin}} > 0I3spin​>0 for all zzz and www. The proof rests on an exact identity: the fermionic and spin reduced density matrices of disjoint regions AAA, DDD separated by BBB differ by the parity insertion (−1)NB(-1)^{N_B}(−1)NB​ in the partial trace. A Perron--Frobenius argument proves element-wise coherence damping; for free fermions, an independent Gaussian bound gives the entropy ordering ΔSAD≥0ΔS_{AD} \ge 0ΔSAD​≥0. Exact diagonalization confirms this for interacting fermions. DMRG on the ttt-VVV chain shows that the factorization contribution exceeds the genuine interaction contribution to I3I_3I3​ by a factor of 8, accounting for ∼80%\sim 80\%∼80% of the deviation in spin-basis numerics. Strong repulsion (K≲0.7K \lesssim 0.7K≲0.7) restores monogamy. Conversely, Z2Z_2Z2​ parity superselection enforces I3≤0I_3 \le 0I3​≤0 at all fillings (proved for w≤3w \le 3w≤3), with the ratio of parity entropy to quantum excess approaching 2ln⁡2/(3ln⁡(4/3))=1.6062\ln 2/(3\ln(4/3)) = 1.6062ln2/(3ln(4/3))=1.606. Any use of I3I_3I3​ as a diagnostic for holographic duality, quantum chaos, or Fermi surface topology must specify the operator algebra; without this, the sign of I3I_3I3​ is ambiguous.