论文标题
纠缠重新归一化和远程纠缠状态的等级
Hierarchy of Entanglement Renormalization and Long-Range Entangled States
论文作者
论文摘要
作为量子性多体物理学的量子信息窗口，纠缠重新归一化组（ERG）的概念和应用在对物质的新量子阶段的研究中起着至关重要的作用，尤其是远程纠缠（LRE）在拓扑秩序的系统中。例如，通过递归应用本地单位以及添加形成产品状态的量子位，第二曲面代码接地状态，即Z_2拓扑顺序的固定点，相对于系统大小有效粗糙。作为进一步的改进，将2D旋转代码的添加/去除到3D X-Cube模型的基础状态中，被证明是必不可少的，并且非常明显地导致了明确定义的大型分裂订单的固定点，这些固定点是非液体样的。在这里，我们提出了一个基本统一的ERG框架，其中允许将一般的自由度递归添加/删除。具体而言，我们在Pauli稳定器代码中建立了ERG和LRE状态的异国情调层次结构，其中2D复合代码和3D X-Cube模型自然包括在内。在层次结构中，可以在更复杂的LRE状态的ERG过程中添加/删除LRE状态，例如3D X-Cube和3D旋转代码接地状态。这样，将大量的Pauli稳定器代码分为一系列``州塔''；对于每座塔，除包括CNOT门在内的本地单位外，较低的LRE级别级别为$ n $，在高级LRE水平状态的级别 - $ n $ erg过程中被添加/删除 - $（n+1）$，连接不同级别的LRE状态和LRE州之间的不同级别的LRE状态。作为未来的指示，我们希望该层次结构可以应用于更一般的LRE国家，从而导致LRE状态的统一ERG场景和精确的Tensor-NetWork表示形式，其形式更为概括。
As a quantum-informative window into quantum many-body physics, the concept and application of entanglement renormalization group (ERG) have been playing a vital role in the study of novel quantum phases of matter, especially long-range entangled (LRE) states in topologically ordered systems. For instance, by recursively applying local unitaries as well as adding/removing qubits that form product states, the 2D toric code ground states, i.e., fixed point of Z_2 topological order, are efficiently coarse-grained with respect to the system size. As a further improvement, the addition/removal of 2D toric codes into/from the ground states of the 3D X-cube model, is shown to be indispensable and remarkably leads to well-defined fixed points of a large class of fracton orders that are non-liquid-like. Here, we present a substantially unified ERG framework in which general degrees of freedom are allowed to be recursively added/removed. Specifically, we establish an exotic hierarchy of ERG and LRE states in Pauli stabilizer codes, where the 2D toric code and 3D X-cube models are naturally included. In the hierarchy, LRE states like 3D X-cube and 3D toric code ground states can be added/removed in ERG processes of more complex LRE states. In this way, a large group of Pauli stabilizer codes are categorized into a series of ``state towers''; with each tower, in addition to local unitaries including CNOT gates, lower LRE states of level-$n$ are added/removed in the level-$n$ ERG process of an upper LRE state of level-$(n+1)$, connecting LRE states of different levels and unveiling complex relations among LRE states. As future directions, we expect this hierarchy can be applied to more general LRE states, leading to a unified ERG scenario of LRE states and exact tensor-network representations in the form of more generalized branching MERA.
