Noncrystalline topological superconductors

Topological insulators, featuring bulk-boundary correspondence, have been realized on a large number of noncrystalline materials, among which amorphous network, quasicrystals, and fractal lattices are the most prominent ones. By contrast, topological superconductors beyond the realm of quantum crystals are yet to be harnessed, as their nucleation takes place around a well-defined Fermi surface with a Fermi momentum, the existence of which rests on the underlying translational symmetry. Here we identify a family of noncrystalline Dirac materials, devoid of time-reversal ($T$) and translational symmetries, on which a suitable local or on-site pairing yields topological superconductors. We showcase this outcome on all the abovementioned noncrystalline platforms embedded in a two-dimensional flat space. The resulting noncrystalline topological superconductors possess quantized topological invariants (Bott index and local Chern marker) and harbor robust one-dimensional Majorana edge modes, analogs of $T$-odd $p+ip$ pairing in noncrystalline materials.