Lower Bound to the Entanglement Entropy of the XXZ Spin Ring

We study the free XXZ quantum spin model defined on a ring of size L and show that the bipartite entanglement entropy of certain eigenstates belonging to the first energy band above the vacuum ground state satisfies a logarithmically corrected area law. This applies in particular to eigenstates corresponding to the lowest eigenenergy above the ground state. To this end, we develop a new perturbational approach, which allows us to control the eigenvalues of reduced states in the XXZ model in terms of the corresponding reduced states in the Ising model. Along the way, we show a Combes–Thomas estimate for fiber operators which can also be applied to discrete many-particle Schrödinger operators on more general translation-invariant graphs.