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Abstract

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.