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Hierarchy of Entanglement Renormalization and Long-Range Entangled States

Meng-Yuan LiPeng Ye
Nov 2022
As a quantum-informative window into condensed matter physics, the conceptand application of entanglement renormalization group (ERG) have been playing avital role in the study of novel quantum phases of matter, especiallytopologically ordered phases that exhibit long range entanglement (LRE)patterns. For instance, by recursively applying local unitaries as well asadding/removing qubits that form product states, the 2D toric code groundstates, i.e., fixed point of $\mathbb{Z}_2$ topological order, are efficientlycoarse-grained with respect to the system size. Recently, in an improved ERGframework, the addition/removal of 2D toric codes into /from the ground statesof the 3D X-cube model, is shown to be indispensable and remarkably leads towell-defined fixed points of a large class of fracton orders. Here, we findthat such progress of ERG is not the end of story if more general degrees offreedom are allowed to be added/removed. Specifically, we establish aninteresting hierarchy structure of ERG and LRE states in a class of Paulistabilizer codes, where the 2D toric code and 3D X-cube model are naturallyincluded. In the hierarchy, LRE states like both 3D X-cube and 3D toric codeground states can be added/removed in an ERG process of more complex LREstates. In this way, all Pauli stabilizer codes considered here are categorizedinto a series of ``state towers''; in each tower, lower LRE states of level-$n$are added/removed in the level-$n$ ERG process of an upper LRE state oflevel-$(n+1)$, which builds bridges between LRE states of different levels. Asfuture directions, we expect this hierarchy can be further generalized in moreexactly solvable models. We also expect the resulting fixed points may admittensor-network representation in the form of more generalized exact branchingMERA.