Defect bulk-boundary correspondence of topological skyrmion phases of matter

Unpaired Majorana zero modes are central to topological quantum computation schemes as building blocks of topological qubits, and are therefore under intense experimental and theoretical investigation. Their gener-alizations to parafermions and Fibonacci anyons are also of great interest, in particular for universal quantum computation schemes. In this work, we find a different generalization of Majorana zero modes in effectively noninteracting systems, which are zero-energy bound states that exhibit a cross structure (two straight, perpen-dicular lines in the complex plane) composed of the complex number entries of the zero-mode wave function on a lattice, rather than a single straight line formed by complex number entries of the wave function on a lattice as in the case of an unpaired Majorana zero mode. These "cross" zero modes are realized for topological skyrmion phases under certain open boundary conditions when their characteristic momentum-space spin textures trap topological defects. They therefore serve as a second type of bulk-boundary correspondence for the topological skyrmion phases. In the process of characterizing this defect bulk-boundary correspondence, we develop recipes for constructing physically relevant model Hamiltonians for topological skyrmion phases, efficient methods for computing the skyrmion number, and introduce three-dimensional topological skyrmion phases into the literature.