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arXiv:2603.08991

Rényi exponent landscape of multipartite entanglement in free-fermion systems

Aleksandrs Sokolovs

cond-mat.stat-mechquant-phSubmitted Mar 9, 2026• Announced Mar 11, 2026• 0 citations
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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.

Comments
6 pages, 1 figure, 1 table

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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 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.
The scaling of entanglement with subsystem size encodes key information about phases and criticality, but the von Neumann entropy is costly to access in experiments and simulations, often requiring full state tomography. The second Rényi entropy is readily measured using two-copy protocols and is often used as a proxy for the von Neumann entanglement entropy, where it is assumed to track its asymptotic scaling. Sugino and Korepiny (Int. J. Mod. Phys. B 32, 1850306 (2018)) revealed that in the ground state of some highly constrained spin models, the scaling of the von Neumann and Rényi entropies can differ, varying from power law to logarithmic scaling as a function of the Rényi index. Here, we construct a number-conserving many-body state that demonstrates a Rényi-index-dependent change in the leading entanglement scaling, generalizing previous results to the case of interacting fermions. We introduce a symmetry-aware lower bound on the von Neumann entropy built from charge-resolved Rényi entropies that can provide a protocol for diagnosing anomalous entanglement scaling from experimentally accessible data.
In a remarkable paper [Phys. Rev. Lett. 96, 100503 (2006)], Gioev and Klich conjectured an explicit formula for the leading asymptotic growth of the spatially bipartite von Neumann entanglement entropy of noninteracting fermions in multidimensional Euclidean space at zero temperature. Based on recent progress by one of us (A. V. S.) in semiclassical functional calculus for pseudodifferential operators with discontinuous symbols, we provide here a complete proof of that formula and of its generalization to Rényi entropies of all orders α>0. The special case α=1/2 is also known under the name logarithmic negativity and often considered to be a particularly useful quantification of entanglement. These formulas exhibiting a "logarithmically enhanced area law" have been used already in many publications.
We uncover a "junction law" for genuine multipartite entanglement, suggesting that in gapped local systems multipartite entanglement is controlled and effectively localized near junctions where subsystem boundaries meet. Using the Rényi-2 genuine multi-entropy GM2(q)\mathrm{GM}^{(\mathtt{q})}_2GM2(q)​ as a diagnostic of genuine q\mathtt{q}q-partite entanglement, we establish this behavior in (2+1)(2+1)(2+1)-dimensional gapped free-fermion lattices with correlation length ξξξ. For partitions with a single junction, GM2(q)\mathrm{GM}^{(\mathtt{q})}_2GM2(q)​ exhibits a universal scaling crossover in L/ξL/ξL/ξ, growing for L≪ξL\llξL≪ξ and saturating to a ξξξ-dependent constant for L≫ξL\ggξL≫ξ, up to O(e−L/ξ)\mathcal{O}(e^{-L/ξ})O(e−L/ξ) corrections. In sharp contrast, for partitions without a junction, GM2(q)\mathrm{GM}^{(\mathtt{q})}_2GM2(q)​ is exponentially suppressed in L/ξL/ξL/ξ and drops below numerical resolution once L≫ξL\ggξL≫ξ. We observe the same pattern for q=3\mathtt{q}=3q=3 (tripartite) and q=4\mathtt{q}=4q=4 (quadripartite) cases, and further corroborate this localization by translating the junction at fixed system size. We also provide a geometric explanation of the junction law in holography. Altogether, these results show that in this gapped free-fermion setting genuine multipartite entanglement is localized within a correlation-length neighborhood of junctions.
At a quantum critical point, bipartite entanglement entropies have universal quantities which are subleading to the ubiquitous area law. For Renyi entropies, these terms are known to be similar to the von Neumann entropy, while being much more amenable to numerical and even experimental measurement. We show here that when calculating universal properties of Renyi entropies, it is important to account for unusual corrections to scaling that arise from relevant local operators present at the conical singularity in the multisheeted Riemann surface. These corrections grow in importance with increasing Renyi index. We present studies of Renyi correlation functions in the 1+11+11+1 transverse-field Ising model (TFIM) using conformal field theory, mapping to free fermions, and series expansions, and the logarithmic entropy singularity at a corner in 2+12+12+1 for both free bosonic field theory and the TFIM, using numerical linked cluster expansions. In all numerical studies, accurate results are only obtained when unusual corrections to scaling are taken into account. In the worst case, an analysis ignoring these corrections can get qualitatively incorrect answers, such as predicting a decrease in critical exponents with the Renyi index, when they are actually increasing. We discuss a two-step extrapolation procedure that can be used to account for the unusual corrections to scaling.
Rényi multi-entropy of graph states and the n−>1n -> 1n−>1 limit
Rényi multi-entropy extends the Rényi entropy to pure states of more than two parties. It is defined by a replica contraction at integer replica number nnn, and the multi-entropy is its n→1n\to1n→1 limit. In holography this limit is conjectured to compute the area of a minimal soap-film web in the bulk. Understanding multi-entropy for general states, however, is harder. Unlike the bipartite case, the replica contraction is not a moment of a density matrix, and the continuation to n=1n=1n=1 has to be found by other means. We study this problem for qubit graph states. We show that at every odd nnn the contraction equals 2−k/22^{-k/2}2−k/2, where kkk is a sum of ranks of matrices over a finite field, one row and one column per qubit. We define a class of graphs that we call collision-controlled, for which the rank sum is a polynomial in nnn. This class includes graphs without even cycles and all complete bipartite graphs. The natural polynomial n→1n \to 1n→1 continuation is a finite sum over partitions of the parties. While the Rényi multi-entropy is non-increasing under single-qubit Pauli measurements at every odd integer n>1n>1n>1, the continuation to n=1n=1n=1 is not. A ZZZ measurement on one qubit of an odd ring with 2r+1≥72r + 1 \geq 72r+1≥7 qubits raises it from (r+1)log⁡2(r+1)\log 2(r+1)log2 to (2r−1)log⁡2(2r-1) \log 2(2r−1)log2. Thus the continuation is not an entanglement monotone.