We consider random translation-invariant frustration-free quantum spin Hamiltonians on $\mathbb Z^D$ in which the nearest-neighbor interaction in every direction is randomly sampled and then distributed across the lattice. Our main result is that, under a small rank constraint, the Hamiltonians are automatically frustration-free and they are gapped with a positive probability. This extends previous results on 1D spin chains to all dimensions. The argument additionally controls the local gap. As an application, we obtain a 2D area law for a cut-dependent ground state via recent AGSP methods of Anshu-Arad-Gosset.

中文翻译：

---

随机平移不变哈密顿量及其谱间隙

我们在 $\mathbb Z^D$ 上考虑随机平移不变的无挫折量子自旋哈密顿量，其中每个方向上的最近邻相互作用被随机采样，然后分布在晶格中。我们的主要结果是，在小秩约束下，哈密顿量自动无挫折，并且它们之间的差距为正概率。这将先前关于一维自旋链的结果扩展到所有维度。该参数还控制局部间隙。作为应用，我们通过 Anshu-Arad-Gosset 的最新 AGSP 方法获得了切割相关基态的 2D 面积定律。