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 \(K-1\).

中文翻译：

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XXZ量子自旋链中的纠缠界

我们考虑以各向异性参数\（\ Delta> 1 \）为特征的XXZ自旋链，并对其进行归一化，以使基态能量为0，基态由全自旋状态给出。能量\（E_K = K（1-1 / \ Delta）\），\（K = 1,2，\ ldots \）可以解释为向下旋转配置的K群集分解阈值。我们证明，对于每一个K，能量低于\（（K + 1）\）-簇破裂的所有状态的二分纠缠都满足对数校正的面积定律。Beaud和Warzel的结果得到了概括，他们考虑了液滴光谱中的能量（即低于2团簇破裂）。对于一般的K，我们发现了一个与因数\（2K-1 \）的对数上限。我们证明此常数在Ising极限\（\ Delta = \ infty \）中是最佳的。Beaud和Warzel还表明，在引入随机场和无序之后，对数校正后的面积定律平均会变成严格的面积定律，对于液滴状态中的状态同样如此。对于具有随机场的伊辛极限，我们证明了该结果不会超出液滴状态。取而代之的是，我们找到能量在K簇破裂上方任意少量的状态，其纠缠满足对数\（K-1 \）的对数增长下界。