Bounds on the entanglement entropy of droplet states in the XXZ spin chain

We consider a class of one-dimensional quantum spin systems on the finite lattice $\mathrm{\Lambda }\subset \mathbb{Z}$, related to the XXZ spin chain in its Ising phase. It includes in particular the so-called droplet Hamiltonian. The entanglement entropy of energetically low-lying states over a bipartition $\mathrm{\Lambda }=B\cup B^c$ is investigated and proven to satisfy a logarithmic bound in terms of $\min \{n,\mathrm{\mid }B\mathrm{\mid },\mathrm{\mid }{B}^c\mathrm{\mid }\}$, where $n$ denotes the maximal number of down spins in the considered state. Upon addition of any (positive) random potential the bound becomes uniformly constant on average, thereby establishing an area law. The proof is based on spectral methods: a deterministic bound on the local (many-body integrated) density of states is derived from an energetically motivated Combes–Thomas estimate.