Aperiodic Weak Topological Superconductors

Abstract

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 non-trivial value of this index implies a non-trivial 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.Introduction -One of the hallmarks of a topological phase is the presence of quantized macroscopic observables which are insensitive to perturbations or random modulations of the local microscopic environment [1, 2]. This remarkable feature can be understood by expressing the quantized observables as topological invariants of an underlying microscopic theory of the bulk system, such that they are unchanged by arbitrary local deformations which do not close the bulk gap. The most well-known example is the transverse conductivity of the quantum Hall effect [3]: owing to the topological protection [4], it shows a level of quantization which is used to define the metrological standard of resistance [5].

Topological Insulators in Amorphous Systems

“…Physics of topological phases in noncrystalline lattices have received relatively less attention. Most studies have focused on quasicrystalline systems [33][34][35] , for example, realizing a weak topological insulator phase in such a system [34]. An interesting unexplored question, both from theoretical and practical perspectives, is whether a completely random set of points i. e., a random lattice, such as that realized by impurities in a material, can host topological phases.…”

Space group constraints on weak indices in topological insulators

“…Weak indices are also present in other symmetry classes [14,16,29,58], and our considerations can be extended to topological superconductors. In 3D there are analogous 2D Z and 2Z indices in classes D and C, respectively; these are Chern numbers of the Bogoliubov-de Gennes (BdG) Hamiltonians, and the same reasoning applies as in class A detailed in Sec.…”

“…As we show below, this constraint holds generally for interacting systems, as long as the ground state is a nonfractionalized 3D insulator which preserves the screw symmetry. We also expect that the constraint remains valid in disordered systems that do not break the symmetry on average [8][9][10]16].…”

“…The lower-dimensional topological invariants are therefore known as weak indices. However, further efforts then showed that weak topological phases have robust topological surface states even in the presence of impurities [8][9][10][11][12], and lattice dislocations therein host protected gapless modes that originate from the weak indices [13][14][15][16]. Recently it was also proposed that strong interactions can lead to novel topological orders on the surface of weak TIs [17][18][19].…”

Real-space mapping of topological invariants using artificial neural networks

“…A more controlled situation is the proposals for topological superconductivity involving nanowires, where the topological state is controlled locally by electric gates [14,15]. Even though real-space formulations for the topological invariant do exist [16][17][18][19][20], their computation requires an integration over the whole space. Thus, there is not a simple methodology to obtain a topological invariant in inhomogeneous systems by evaluating solely their local properties.…”