Entanglement, self-similarity and emergent quantum order in fractal lattice systems

Motivated by recent experimental advances in realizing quantum matters with fractal geometry and in fractional dimension, we study the entanglement of many-qudit systems on fractal lattices and the quantum order emerging from thereon. In such systems, translational symmetry has no place while the fractal self-similarity shapes the scaling, hence indicating novel quantum phase of matter. We try to explore such novelty with Bose-Hubbard-type models whose ground states are entanglement-renormalization fixed points with long-range entanglement patterns. We show that a quantum-error-correction structure capturing the low-energy physics gives rise to the self-similar and long-range entanglement pattern, and portrays the emergent quantum order. We also show how numerical tools can be used to probe such novel quantum phases.