Entanglement and Majorana edge states in the Kitaev model

Abstract

We investigate the von Neumann entanglement entropy and Schmidt gap in the vortex-free ground state of the Kitaev model on the honeycomb lattice for square/rectangular and cylindrical subsystems. We find that, for both the subsystems, the free-fermionic contribution to the entanglement entropy ${S}_{E}$ exhibits signatures of the phase transitions between the gapless and gapped phases. However, within the gapless phase, we find that ${S}_{E}$ does not show an expected monotonic behavior as a function of the coupling ${J}_{z}$ between the suitably defined one-dimensional chains for either geometry; moreover, the system generically reaches a point of minimum entanglement within the gapless phase before the entanglement saturates or increases again until the gapped phase is reached. This may be attributed to the onset of gapless modes in the bulk spectrum and the competition between the correlation functions along various bonds. In the gapped phase, on the other hand, ${S}_{E}$ always monotonically varies with ${J}_{z}$ independent of the subregion size or shape. Finally, further confirming the Li-Haldane conjecture, we find that the Schmidt gap $\mathrm{\ensuremath{\Delta}}$ defined from the entanglement spectrum also signals the topological transitions but only if there are corresponding zero-energy Majorana edge states that simultaneously appear or disappear across the transitions. We analytically corroborate some of our results on entanglement entropy, the Schmidt gap, and the bulk-edge correspondence using perturbation theory.