Perfectly localized Majorana corner modes in fermionic lattices

The conducting bulk and surface modes of topological systems can be gapped out, resulting in systems which are commonly referred to as higher order topological insulators (HOTIs). Focusing on examples of Majorana zero modes on the corners of a two-dimensional system, we introduce a method to find parameter regions where the Majorana modes are perfectly localised on a single site, with zero correlation length. Such a limit allows us to study the dimerisation structure of the sparse bulk Hamiltonian that results in the higher order topology of the system. Furthermore, such limits typically feature a feasible analytical understanding of the system's energy scales. Based on the dimerisation structure we extract from the 2D model, we identify a more general stacking procedure to construct Majorana zero modes in arbitrary corners of a $d$ - dimensional hypercube, which we demonstrate explicitly in 3D. Our construction requires no inherent crystalline symmetries, and the Majorana corner modes are protected by the particle-hole symmetry of edge Hamiltonians, inherited from the BdG Hamiltonians of the bulk. Edited