Entanglement Bounds in the XXZ Quantum Spin Chain

We consider the XXZ spin chain, characterized by an anisotropy parameter $$\Delta >1$$ , and normalized such that the ground state energy is 0 and the ground state given by the all-spins-up state. The energies $$E_K = K(1-1/\Delta )$$ , $$K=1,2,\ldots $$ , can be interpreted as K-cluster breakup thresholds for down-spin configurations. We show that, for every K, the bipartite entanglement of all states with energy below the $$(K+1)$$ -cluster breakup satisfies a log-corrected area law. This generalizes a result by Beaud and Warzel, who considered energies in the droplet spectrum (i.e., below the 2-cluster breakup). For general K, we find an upper logarithmic bound with pre-factor $$2K-1$$ . We show that this constant is optimal in the Ising limit $$\Delta =\infty $$ . Beaud and Warzel also showed that after introducing a random field and disorder averaging the log-corrected area law becomes a strict area law, again for states in the droplet regime. For the Ising limit with random field, we show that this result does not extend beyond the droplet regime. Instead, we find states with energies of an arbitrarily small amount above the K-cluster breakup whose entanglement satisfies a logarithmically growing lower bound with pre-factor $$K-1$$ .