Entanglement and Majorana edge states in the Kitaev model

TL;DR

This paper studies entanglement entropy and edge states in the Kitaev model, revealing how they signal phase transitions and topological changes, with analytical support for the observed phenomena.

Contribution

It provides new insights into entanglement behavior and edge state signatures in the Kitaev model, especially regarding the non-monotonic entanglement evolution within the gapless phase.

Findings

Entanglement entropy signals phase transitions but shows non-monotonic behavior in the gapless phase.

Schmidt gap indicates topological transitions only when Majorana edge states are present.

Analytical results support the connection between entanglement measures and topological features.

Abstract

We investigate the von Neumann entanglement entropy and Schmidt gap in the ground state of the Kitaev model on the honeycomb lattice for square 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, we find that $S_E$ does not show an expected monotonic behaviour as a function of the coupling $J_z$ between the suitably defined one-dimensional chains within the gapless phase 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 competition between the nearest neighbour correlation functions of the $x-x$ and $z-z$ bonds, and to the onset of gapless $\vec{k}$-modes in the bulk spectrum. On the other hand $S_E$ always monotonically varies with $J_z$ in the gapped phase independent of the subregion size or shape. Finally, further confirming the Li-Haldane conjecture, we find that the Schmidt gap $\Delta$ 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, Schmidt gap and the bulk-edge correspondence using perturbation theory.