Theory of collective topologically protected Majorana fermion excitations of networks of localized Majorana modes

Predictions of localized Majorana modes, and ideas for manipulating these degrees freedom, are the two key ingredients in proposals physical platforms quantum computation. Several envisage a scalable network such modes coupled bilinearly to each other by quantum-mechanical mixing amplitudes. Here, we develop theoretical framework characterizing collective topologically protected zero-energy fermion excitations networks modes. A ingredient our work is Gallai-Edmonds decomposition general graph, which use obtain an alternate ``local'' proof ``global'' result Lov\'asz Anderson on dimension null space real skew-symmetric (or pure-imaginary Hermitian) adjacency matrices graphs. Our approach Anderson's constructs maximally basis said from graph. Applied graph question, this gives method basis-independent properties excitations, relating correlation function monomers ensemble maximum matchings (maximally packed dimer covers) corresponding can also be used identify signatures systems modeled free-fermion Hamiltonian with hopping matrix type; interesting example provided vacancy-induced Curie tails generalizations (on nonbipartite lattices) Kitaev's honeycomb model.