Multipartite entanglement in the random Ising chain

Abstract

Quantifying entanglement of multiple subsystems is a challenging open problem in interacting quantum systems. Here, we focus on two subsystems of length $\ensuremath{\ell}$ separated by a distance $r=\ensuremath{\alpha}\ensuremath{\ell}$ and quantify their entanglement negativity ($\mathcal{E}$) and mutual information ($\mathcal{I}$) in critical random Ising chains. We find universal constant $\mathcal{E}(\ensuremath{\alpha})$ and $\mathcal{I}(\ensuremath{\alpha})$ over any distances, using the asymptotically exact strong disorder renormalization group method. Our results are qualitatively different from both those in the clean Ising model and random spin chains of a singlet ground state, like the spin-$\frac{1}{2}$ random Heisenberg chain and the random XX chain. While for random singlet states $\mathcal{I}(\ensuremath{\alpha})/\mathcal{E}(\ensuremath{\alpha})=2$, in the random Ising chain this universal ratio is strongly $\ensuremath{\alpha}$ dependent. This deviation between systems contrasts with the behavior of the entanglement entropy of a single subsystem, for which the various random critical chains and clean models give the same qualitative behavior. The reason is that $\mathcal{E}$ and $\mathcal{I}$ are sensitive to higher order correlations in the ground-state structure. Therefore, studying multipartite entanglement provides additional universal information in random quantum systems, beyond what we can learn from a single subsystem.