# Entanglement bounds in the XXZ quantum spin chain

## TL;DR

This paper investigates entanglement properties in the XXZ quantum spin chain, establishing bounds on entanglement entropy for states below certain energy thresholds, and explores effects of disorder and anisotropy on these bounds.

## Contribution

It generalizes previous results by providing entanglement bounds for all cluster break-up thresholds in the XXZ chain, including the effects of disorder and the Ising limit.

## Key findings

- Logarithmically corrected area law for entanglement in states below the (K+1)-cluster break-up energy.
- Optimality of the logarithmic bound in the Ising limit.
- Existence of states with near-threshold energies exhibiting lower bound growth in entanglement.

## Abstract

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 break-up thresholds for down spin configurations. We show that, for every $K$, the bipartite entanglement of all states with energy below the $(K+1)$-cluster break-up satisfies a logarithmically corrected (or enhanced) area law. This generalizes a result by Beaud and Warzel, who considered energies in the droplet spectrum (i.e., below the 2-cluster break-up). 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 enhanced 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 arbitrarily close to the $(K+1)$-cluster break-up whose entanglement satisfies a logarithmically growing lower bound with pre-factor $K-1$.