Beyond Area Law: Network-Irreducible Multiparty Entanglement in Quantum Matter

Collective long-range entanglement is a defining feature of quantum phases and a key to identifying intrinsically new orders such as topological phases. A rigorous measure of collective quantum correlations, the Genuine Multipartite Entanglement (GME), has been widely used to probe quantum criticality, and more recently, as a way to characterize exotic phases such as the quantum spin liquids. Here, we show that GME contains a trivial area-law contribution originating from bipartite bonds, which masks the true collective entanglement among subsystems. To isolate this irreducible component, we develop new tools to quantify the Genuine Network Multipartite Entanglement (GNME), recently introduced in the context of quantum networks. We benchmark GNME on canonical few-qubit states, providing the first rigorous upper bounds on their white-noise robustness, and then apply it to quantum matter. In the 1d quantum Ising model, GNME serves as a sharper indicator of quantum criticality, decaying much faster than GME as the field is tuned or the temperature is increased away from the critical point. In contrast, some reduced states of spin-liquid phases exhibit strong GME but weak or vanishing GNME, revealing that their entanglement structure is primarily bipartite in origin. Our results establish GNME as a new lens to distinguish truly collective entanglement from short-range correlations in quantum materials.