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arXiv:1601.01695 · Physical review. B./Physical review. B

Entanglement and Majorana edge states in the Kitaev model

Saptarshi Mandal, Moitri Maiti, Vipin Kerala Varma

Published Jul 12, 2016• 13 citations
arXivHTMLPDF

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.

DOI
10.1103/physrevb.94.045421

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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 SE{S}_{E}SE​ exhibits signatures of the phase transitions between the gapless and gapped phases. However, within the gapless phase, we find that SE{S}_{E}SE​ does not show an expected monotonic behavior as a function of the coupling Jz{J}_{z}Jz​ 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, SE{S}_{E}SE​ always monotonically varies with Jz{J}_{z}Jz​ independent of the subregion size or shape. Finally, further confirming the Li-Haldane conjecture, we find that the Schmidt gap \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ 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.
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The Kitaev model is exactly solvable in terms of Majorana fermions hopping on a honeycomb lattice and coupled to a static Z2\mathbb{Z}_2Z2​ gauge field, giving the possibility of πππ-vortices in hexagonal plaquettes. In the vortex-full sector and in the presence of a time-reversal-breaking three-spin term of strength κκκ, the energy spectrum is gapped and the ground state possesses an even Chern number. An isolated vortex-free plaquette acts as a ``dual vortex'' and binds a fermionic mode at finite energy εεε in the bulk gap. This mode is equivalent to two coupled Majorana zero modes located on the same dual vortex. In a continuum approximation, we analytically compute the Majorana wavefunctions and their coupling εεε in the two limits of small or large κκκ. The analytical approach is confirmed by numerical perturbation theory directly on the lattice. The latter is in excellent agreement with the full numerics on a finite-size system. We contrast our results with states bound to an isolated vortex in a topological superconductor with even Chern number.
We study the entanglement properties of a three dimensional generalization of the Kitaev honeycomb model proposed by Ryu [Phys. Rev. B 79, 075124, (2009)]. The entanglement entropy in this model separates into a contribution from a Z2Z_2Z2​ gauge field and that of a system of hopping Majorana fermions, similar to what occurs in the Kitaev model. This separation enables the systematic study of the entanglement of this 3D interacting bosonic model by using the tools of non-interacting fermions. In this way, we find that the topological entanglement entropy comes exclusively from the Z2Z_2Z2​ gauge field, and that it is the same for all of the phases of the system. There are differences, however, in the entanglement spectrum of the Majorana fermions that distinguish between the topologically distinct phases of the model. We further point out that the effect of introducing vortex lines in the Z2Z_2Z2​ gauge field will only change the entanglement contribution of the Majorana fermions. We evaluate this contribution to the entanglement which arises due to gapless Majorana modes that are trapped by the vortex lines.
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Entanglement signatures of gapless topological phases in a ppp-wave superconductor
We explore the gapless topological phases of a ppp-wave superconductor, probing its rich topologically ordered phases and underlying quantum phenomena. The topological order of the system is characterized by studying its entanglement properties. This study confirms the bulk-boundary correspondence in the entanglement spectrum, even without a full bulk gap. For contractible bipartitions, the entanglement entropy varies non-monotonically with the chemical potential, displaying pronounced peaks at points where the bulk gap closes and reopens, signaling topological quantum phase transitions. This behavior remains robust in the thermodynamic limit. The entanglement entropy grows with system size for non-contractible bipartitions, indicating long-range entanglement in the gapless phase. These findings reveal the subtle interplay between symmetry, entanglement, and topology in gapless systems, and emphasize the role of entanglement-based diagnostics in identifying unconventional topological phases beyond the gapped paradigm.