The study of low-dimensional quantum systems has proven to be a particularly fertile field for discovering novel types of quantum matter. When studied numerically, low-energy states of low-dimensional quantum systems are often approximated via a tensor-network description. The tensor network's utility in studying short range correlated states in 1D have been thoroughly investigated, with numerous examples where the treatment is essentially exact. Yet, despite the large number of works investigating these networks and their relations to physical models, examples of exact correspondence between the ground state of a quantum critical system and an appropriate scale-invariant tensor network have eluded us so far. Here we show that the features of the quantum-critical Motzkin model can be faithfully captured by an analytic tensor network that exactly represents the ground state of the physical Hamiltonian. In particular, our network offers a two-dimensional representation of this state by a correspondence between walks and a type of tiling of a square lattice. We discuss connections to renormalization and holography.

中文翻译：

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Motzkin 自旋链的精确全息张量网络

低维量子系统的研究已被证明是发现新型量子物质的一个特别肥沃的领域。当进行数值研究时，低维量子系统的低能态通常通过张量网络描述来近似。已经彻底研究了张量网络在研究一维短程相关状态中的效用，其中有许多例子表明处理基本上是准确的。然而，尽管有大量工作研究这些网络及其与物理模型的关系，但迄今为止，量子临界系统的基态与适当的尺度不变张量网络之间精确对应的例子仍未得到我们的了解。在这里，我们展示了量子临界 Motzkin 模型的特征可以被精确表示物理哈密顿量基态的解析张量网络忠实地捕获。特别是，我们的网络通过步行和方形格子的平铺类型之间的对应关系提供了这种状态的二维表示。我们讨论与重整化和全息的联系。