While volume violation of area law has been exhibited in several quantum spin chains, the construction of a corresponding ground state in higher dimensions, entangled in more than one direction, has been an open problem. Here we construct a 2D frustration-free Hamiltonian with maximal violation of the area law. We do so by building a quantum model of random surfaces with color degree of freedom that can be viewed as a collection of colored Dyck paths. The Hamiltonian may be viewed as a 2D generalization of the Fredkin spin chain. It relates all the colored random surface configurations subject to a Dirichlet boundary condition and hard wall constraint from below to one another, and the ground state is therefore a superposition of all such classical states and non-degenerate. Its entanglement entropy between subsystems undergoes a quantum phase transition as the deformation parameter is tuned. The area- and volume-law phases are similar to the one-dimensional model, while the critical point scales with the linear size of the system $L$ as $L\log L$. Further it is conjectured that similar models with entanglement phase transitions can be built in higher dimensions with even softer area law violations at the critical point.

中文翻译：

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量子菱形平铺和纠缠相变


虽然体积违反面积定律的情况已经在几个量子自旋链中表现出，但在更高维度上构建相应的基态，在多个方向上纠缠，一直是一个悬而未决的问题。在这里，我们构造了一个 2D 无挫折哈密顿量，最大程度地违反了面积定律。我们通过构建具有颜色自由度的随机表面的量子模型来实现这一点，该模型可以被视为彩色 Dyck 路径的集合。哈密顿量可以看作是 Fredkin 自旋链的 2D 泛化。它将所有受狄利克雷边界条件和硬壁约束约束的彩色随机表面配置从下相互关联，因此基态是所有这些经典态和非简并态的叠加。随着变形参数的调整，它在子系统之间的纠缠熵会经历量子相变。面积定律和体积定律相位类似于一维模型，而临界点随系统 $L$ 的线性大小缩放，如 $L\log L$。此外，可以推测，具有纠缠相变的类似模型可以在更高的维度上构建，在临界点处甚至会违反更软的面积定律。