  Recently, the structural disorder-induced topological phase transitions in
periodic systems have attracted much attention. However, in aperiodic systems
such as quasicrystalline systems, the interplay between structural disorder and
band topology is still unclear. In this work, we investigate the effects of
structural disorder on a quantum spin Hall insulator phase and a higher-order
topological phase in a two-dimensional Amman-Beenker tiling quasicrystalline
lattice, respectively. We demonstrate that the structural disorder can induce a
topological phase transition from a quasicrystalline normal insulator phase to
an amorphous quantum spin Hall insulator phase, which is confirmed by bulk gap
closing and reopening, robust edge states, quantized spin Bott index and
conductance. Furthermore, the structural disorder-induced higher-order
topological phase transition from a quasicrystalline normal insulator phase to
an amorphous higher-order topological phase characterized by quantized
quadrupole moment and topological corner states is also found. More strikingly,
the disorder-induced higher-order topological insulator with eight corner
states represents a distinctive topological state that eludes realization in
conventional crystalline systems. Our work extends the study of the interplay
between disorder effects and topologies to quasicrystalline and amorphous
systems.
