Hierarchy of Entanglement Renormalization and Long-Range Entangled States

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.