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

TL;DR

This paper demonstrates that multi-entropy in random tensor networks cannot exhibit replica symmetry breaking due to structural incompatibilities in boundary permutations, contrasting with negativity which does exhibit RSB.

Contribution

It reveals a fundamental structural obstruction to RSB for multi-entropy in RTN models, contrasting with negativity's RSB behavior, supported by numerical evidence.

Findings

Multi-entropy has an obstruction to RSB for all Re9nyi indices and multipartite numbers.

Boundary permutations for multi-entropy are incompatible with a common geodesic permutation.

Negativity exhibits RSB while multi-entropy does not, confirmed by numerical simulations.

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 R\'enyi 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 $\tau$ in the Cayley graph of $S_N$. This is in sharp contrast to entanglement negativity, which does admit such a $\tau$-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.