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We study the gapped phase of Kitaev's honeycomb model (a $Z_2$ spin liquid)on a lattice with topological defects. We find that some dislocations andstring defects carry unpaired Majorana fermions. Physical excitationsassociated with these defects are (complex) fermion modes made out of two(real) Majorana fermions connected by a $Z_2$ gauge string. The quantum stateof these modes is robust against local noise and can be changed by winding a$Z_2$ vortex around one of the dislocations. The exact solution respects gaugeinvariance and reveals a crucial role of the gauge field in the physics ofMajorana modes. To facilitate these theoretical developments, we recast thedegenerate perturbation theory for spins in the language of Majorana fermions.

Unpaired Majorana modes on dislocations and string defects in Kitaev's honeycomb model