Quantum Phase Transitions in Quasicrystalline Higher-Order Topological Insulators

Topological phase transitions in disordered systems exhibit rich critical behaviors, including the emergence of multifractal wavefunctions. However, topological phase transitions in quasicrystalline systems have received relatively limited attention, despite the rapidly growing research on quasicrystalline topological phases. This study investigates the phase transition between a quasicrystalline higher-order topological insulator (HOTI) and a trivial band insulator. We perform model studies on two distinct systems: an Ammann-Beenker (AB) tiling quasicrystal and a square lattice with a quasiperiodic potential. Our findings reveal common low-energy physics shared by these systems: (i) We observe the presence of a gapless, critical region that separates the HOTI and trivial phases. This critical region is characterized by a high density of states and multifractal wavefunctions near the Fermi level. (ii) The topological phase transition, i.e., jump of fractional corner charge, occurs at a specific point in the critical region where the eigenstates at the Fermi level are maximally extended. Our findings suggest the existence of a novel gapless phase with simultaneous fractional corner charge and multifractal states at the Fermi level, shedding light on intriguing and previously unexplored facets of topological transitions within quasicrystalline systems.