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arXiv:2604.13261

Structural Obstruction to Replica Symmetry Breaking for Multi-Entropy in Random Tensor Networks

Sriram Akella, Norihiro Iizuka

Published Apr 14, 2026• 0 citations
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Abstract

We study replica symmetry breaking (RSB) for multi-entropy in the random-tensor-network (RTN) domain-wall spin model. Our main result is that, within this framework, multi-entropy has a structural obstruction to RSB for any Rényi index $n$ and any multipartite number $\mathtt{q}$. This obstruction arises because the boundary permutations relevant to multi-entropy are organized along mutually incompatible coordinate directions of the replica hypercube, and therefore do not admit a nontrivial common geodesic intermediate permutation $τ$ in the Cayley graph of $S_N$. This is in sharp contrast to entanglement negativity, which does admit such a $τ$-mediated saddle and exhibits RSB in the same framework. As a robustness check, we also consider a toy $\mathbb{Z}_2$ gauge extension of the spin model with a minimal bulk gauge constraint. Numerical evidence in this gauged model indicates that multi-entropy continues to show no sign of RSB at $n=2$ and $n=3$, while negativity continues to exhibit RSB. Our results show that, within the RTN spin-model description, multi-entropy is not "RSB-friendly'': its boundary data are structurally incompatible with a nontrivial common geodesic intermediate permutation, unlike negativity.

DOI
10.48550/arxiv.2604.13261

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We study the evaluation of Rényi multi-entropies Sn(q)S^{(q)}_nSn(q)​ in Random Tensor Network (RTN) states in the large bond-dimension limit. For the case of Rényi index n=2n=2n=2 and arbitrary number of parties qqq, we prove that that multi-entropies are determined by minimal multiway cuts through the network. When the minimal multiway cut is degenerate, we characterize the full minimizer set via compatible families of minimal cuts and give a criterion for all minimizers to come from ordinary cut partitions. For n=2n=2n=2, this gives a natural generalization of the minimal cut description of bipartite entanglement to multipartite systems with arbitrarily many parties. For the case of integer n>2n>2n>2, we show that the minimal multiway cut conjecture is in general \emph{not true} by providing explicit counter examples for both the single random tensor and for the network built from isometric tilings. We discuss the implication for our results on the multipartite entanglement structures in RTN and holography.
For a static time slice of AdS _3 3 we describe a particular class of minimal surfaces which form trivalent networks of geodesics. Through geometric arguments we provide evidence that these surfaces describe a measure of multipartite entanglement. By relating these surfaces to Ryu-Takayanagi surfaces it can be shown that this multipartite contribution is related to the angles of intersection of the bulk geodesics. A proposed boundary dual [Phys. Rev. D 106, 126001 (2022), J. High Energy Phys. 08, 202 (2023), J. High Energy Phys. 05, 008 (2023)], the multi-entropy, generalizes replica trick calculations involving twist operators by considering monodromies with finite group symmetry beyond the cyclic group used for the computation of entanglement entropy. We make progress by providing explicit calculations of Renyi multi-entropy in two dimensional CFTs and geometric descriptions of the replica surfaces for several cases with low genus. We also explore aspects of the free fermion and free scalar CFTs. For the free fermion CFT we examine subtleties in the definition of the twist operators used for the calculation of Renyi multi-entropy. In particular the standard bosonization procedure used for the calculation of the usual entanglement entropy fails and a different treatment is required.
We investigate multipartite entanglement for quantum states of 3d space geometry, described via generalised random spin networks with fixed areas, in the context of background independent approaches to quantum gravity. We focus on entanglement negativity as a well defined witness of quantum correlations for mixed states, in our setting describing generic subregions of the boundary of a quantum 3d region of space. In particular, we consider a generic tripartition of the boundary of an open spin network state and we compute the typical Rényi negativity of two boundary subregions A and B immersed in the environment C, explicitly for a set of simple open random spin network states. We use the random character of the spin network to exploit replica and random average techniques to derive the typical Rényi negativty via a classical generalised Ising model correspondence, generally used for random tensor networks in the large bond regime. For trivially correlated random spin network states, with only local entanglement between spins located on the network edges, we find that typical log negativity displays a holographic character, in agreement with the results for random tensor networks, in large spin limit. When non-local bulk entanglement between intertwiners at the vertices is considered the negativity increases, while at the same time the holographic scaling is generally perturbed by the bulk contribution.
A bstract Since the work of Ryu and Takayanagi, deep connections between quantum entanglement and spacetime geometry have been revealed. The negative eigenvalues of the partial transpose of a bipartite density operator is a useful diagnostic of entanglement. In this paper, we discuss the properties of the associated entanglement negativity and its Rényi generalizations in holographic duality. We first review the definition of the Rényi negativities, which contain the familiar logarithmic negativity as a special case. We then study these quantities in the random tensor network model and rigorously derive their large bond dimension asymptotics. Finally, we study entanglement negativity in holographic theories with a gravity dual, where we find that Rényi negativities are often dominated by bulk solutions that break the replica symmetry. From these replica symmetry breaking solutions, we derive general expressions for Rényi negativities and their special limits including the logarithmic negativity. In fixed-area states, these general expressions simplify dramatically and agree precisely with our results in the random tensor network model. This provides a concrete setting for further studying the implications of replica symmetry breaking in holography.
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.