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 SE exhibits signatures of the phase transitions between the gapless and gapped phases. However within the gapless phase, we find that SE does not show an expected monotonic behaviour as a function of the coupling 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 always monotonically varies with Jz independent of the sub-region size or shape. Finally, further confirming the Li-Haldane conjecture, we find that the Schmidt gap ∆ 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.

Dynamical merging of Dirac points in the periodically driven Kitaev honeycomb model

“…The important point is that through the scheme outlined the phases of the model can be externally controlled and indeed fine tuned phases can be created where the model reduces to an effective Ising model (at the intersection of two merge lines). Finally, we note that merging of DPs involves transition from gappless to gapped phases which may be experimentally captured in the entanglement entropy [70] as also theoretically predicted in [71].…”

Schmidt gap in random spin chains

“…The Schmidt gap, the difference of the two largest Schmidt eigenvalues of the ES, originally introduced in [18] and [19] was shown to scale according to universal critical exponents in [20][21][22][23][24]. It was further employed in the characterization of 2D spin models in a region close to a topological spin liquid [25,26]. The time evolution of the Schmidt gap was analyzed in [27][28][29] for the dynamics after a quantum quench in homogeneous systems and in [30] for a quench to a many-body localized Hamiltonian.…”

Two-rate periodic protocol with dynamics driven through many cycles

“…Moreover, finding the Floquet spectrum and the possible change in its topology, already observed for r = 0 case using a square wave protocol 29 , can also give useful information on different dynamical phases and the transitions between them. Furthermore, we may also look into the entanglement spectrum and Schmidt gap in the dynamically evolved states and probe the possibilities of any topological transition 44 there. In short, the tuning knob r of the two rate periodic protocol opens up a world of opportunities enabling exploration of multifarious dynamical phenomena with huge possibilities for numerous practical implementations.…”