A long-range entangled state beyond the paradigm of topological order

The nature of the famed long-range entanglement of quantum many-body systems is the paradoxical coexistence of short-range correlation and nonlocal information which cannot be encoded through constant-depth local quantum circuits. This coexistence is only proved in topologically ordered states and their extensions in 2D or higher dimensions, and is in charge of their fundamental importance in fault-tolerant quantum computation. Here, we broaden the knowledge of long-range entangled states beyond the manifold-topology picture by proving the coexistence in a new concrete example in addition to those well-known ones like the toric-code ground state. The new long-range entangled state reveals an intrinsically different paradigm, it describes quantum matter on the newly experimentally realized fractal lattice geometry in 1.58D where the topological framework is negated, but a self-similarity structure shapes how the nonlocal information is encoded. The new paradigm portrays an exotic quantum order and uncovers the potential of fractional-dimension quantum states in studies relevant to quantum computation.