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

Abstract

We study the gapped phase of Kitaev's honeycomb model (a Z2 spin liquid) on a lattice with topological defects. We find that some dislocations and string defects carry unpaired Majorana fermions. Physical excitations associated with these defects are (complex) fermion modes made out of two (real) Majorana fermions connected by a Z2 gauge string. The quantum state of these modes is robust against local noise and can be changed by winding a Z2 vortex around one of the dislocations. The exact solution respects gauge invariance and reveals a crucial role of the gauge field in the physics of Majorana modes. To facilitate these theoretical developments, we recast the degenerate perturbation theory for spins in the language of Majorana fermions.

Demonstrating non-Abelian statistics of Majorana fermions using twist defects

“…The choice of the value of the link operator fixes the Z 2 background gauge. As a result, the Hamiltonian becomes quadratic in terms of η Majorana modes and can be solved exactly 5,36 . A pair of twist dislocations can be added by modifying the bonds according to Fig.…”

“…So far, majority of the activity has focused on topological superconductors supporting localized zero-energy modes [18][19][20] . However, it has been shown by Petrova et al 35,36 that introducing topological defects in the Abelian phase of the Kitaev honeycomb model 37 provides an alternative way to generate unpaired Majorana zero modes. Previously, twist defects in the toric code model are shown to exhibit Ising anyon like behavior through either fusion and braiding rules 9-11 or a mean-field treatment 7 .…”

“…Previously, twist defects in the toric code model are shown to exhibit Ising anyon like behavior through either fusion and braiding rules 9-11 or a mean-field treatment 7 . Here, it is our aim to demonstrate explicitly the emergence of unpaired Majorana zero modes associated with the twist defects as has been done in the Kitaev honeycomb model 35,36 . The toric code model was initially introduced by Kitaev 5 and later reformulated by Wen 6 .…”

“…It has been shown that the Kitaev honeycomb model admits unpaired Majorana modes after introducing dislocations into the lattice 35,36 . Naturally, one would expect the unpaired Majorana zero modes associated with twist defects in the toric code model can be deduced from the Kitaev honeycomb model with a certain type of dislocations.…”

Continuous error correction for Ising anyons

“…[8][9][10][11] for reviews. Ising anyons also appear as excitations [12][13][14][15] or ends of defect lines [16][17][18][19] in several spin-lattice models.…”