We consider the XXZ chain in the Ising phase. The particle number conservation property is used to write the Hamiltonian in a hard-core particles formulation over the $N$-symmetric product of graphs, where $N\in\mathbb{N}_0$ is the number of conserved particle. The droplet regime corresponds to a band at the bottom of the spectrum of the model consisting of a connected set (a droplet) of down-spins, up to an exponential error. It is interesting to know that in the formulation over the $N$-symmetric product graphs, with a fixed $N\geq 1$, the XXZ chain can be seen as a one-dimensional model only when it is restricted to droplet states. This justifies the recent many-body localization indicators proved in the droplet regime by Elgart/Klein/Stolz and Beaud/Warzel for the disordered model, including an area law of arbitrary states in that localized phase. As a first step beyond the droplet regime, we show that the entanglement of arbitrary states above the droplet regime (associated with multiple droplets/clusters) does not follow area laws, and instead, it follows a logarithmically corrected (enhanced) area law. We will comment on the effects of disorder on entanglement, and show how our results hint a phase transition.
(joint work with C. Fischbacher and G. Stolz, arXiv1907.11420)