Aperiodic Weak Topological Superconductors

Weak topological phases are usually described in terms of protection by the lattice translation symmetry. Their characterization explicitly relies on periodicity since weak invariants are expressed in terms of the momentum-space torus. We prove the compatibility of weak topological superconductors with aperiodic systems, such as quasicrystals. We go beyond usual descriptions of weak topological phases and introduce a novel, real-space formulation of the weak invariant, based on the Clifford pseudospectrum. A nontrivial value of this index implies a nontrivial bulk phase, which is robust against disorder and hosts localized zero-energy modes at the edge. Our recipe for determining the weak invariant is directly applicable to any finite-sized system, including disordered lattice models. This direct method enables a quantitative analysis of the level of disorder the topological protection can withstand.

Figure 1

Left: Patch of the Ammann-Beenker tiling in the $x\text{−}y$ plane, obtained by repeated subdivision of a square [45]. A tight binding model $H_{\mathrm{QC}}$ is defined by associating an on-site Hamiltonian and a hopping matrix to each vertex and link in the patch, respectively. Right: Total wave function amplitude of $H_{\mathrm{QC}}$, corresponding to states with energies $|E|<0.2$, for $t=\mathrm{\Delta }=1$ and $\mu =2$. Circles of larger area and darker color correspond to larger amplitudes.

Figure 2

Log-linear plot of the thermal conductance distribution of a single edge in the WTI phase. The histogram (solid line) is computed numerically from $2\times {10}^5$ disorder realizations, using $t=\mathrm{\Delta }=U=1$ and $\mu =1.9$, for a quasicrystal patch composed of 13 weakly coupled wires. The dashed line shows the analytic result of Eq. (9), using $l/L=0.18$. Inset: Total amplitude of wave functions with energies $|E|<0.1$ for a single disorder realization. Circles of larger area and darker color correspond to larger amplitudes, while thicker hoppings show the positions of Kitaev chains in the array.