Robustness of topological order against disorder

A universal topological marker has been proposed recently to map the topological invariant of Dirac models in any dimension and symmetry class to lattice sites. Using this topological marker, we examine the conditions under which the global topological order, represented by the average topological marker, remains unchanged in the presence of disorder for 1D, 2D and 3D systems. We find that if an impurity corresponds to varying a nonzero matrix element of the lattice Hamiltonian, regardless the element represents hopping, chemical potential, pairing, etc, then the average topological marker is conserved. However, if there are many strong impurities and the average distance between them is shorter than a correlation length, then the average marker is no longer conserved. In addition, strong and dense impurities can be used to continuously interpolate between one topological phase and another. A number of prototype lattice models including Su-Schrieffer-Heeger model, Majorana chain, Chern insulators, Bernevig-Hughes-Zhang model, chiral $p$-wave superconductors, and 3D topological insulators are used to elaborate the ubiquity of these statements.