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Physical Review B

Dynamics of the Kitaev chain model under parametric pumping

Andrei A. Zvyagin

Published Jul 18, 2014• 11 citations
Publisher

Abstract

Dynamics of the Kitaev chain model under the effect of parametric pumping is studied. Two contributions to dynamical characteristics are considered: from the extended eigenstates and from the edge bound state (zero Majorana modes). It is shown that in the dynamical regime the frequencies of Rabi oscillations for zero Majorana modes are much larger than those related to gapped extended states. In the steady-state regime, the Rabi oscillations are blurred due to relaxation processes, and only oscillations of the characteristics of the model with the pumping frequency exist, producing absorption of the pumping power by extended states. Experimental realizations of the considered effect are discussed.

DOI
10.1103/physrevb.90.014507

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Dynamics of the Kitaev chain model under the effect of parametric pumping is studied. Two contributions to dynamical characteristics are considered: from the extended eigenstates and from the edge bound state (zero Majorana modes). It is shown that in the dynamical regime the frequencies of Rabi oscillations for zero Majorana modes are much larger than those related to gapped extended states. In the steady-state regime, the Rabi oscillations are blurred due to relaxation processes, and only oscillations of the characteristics of the model with the pumping frequency exist, producing absorption of the pumping power by extended states. Experimental realizations of the considered effect are discussed.
With the help of the exact solution of the Kitaev model the parametric pumping of the two-dimensional quantum spin liquid under the action of the ac magnetic field is studied. In the dynamical regime the field produces oscillations of the magnetization with the field's frequency, modulated by the Rabi-like oscillations. In the steady-state regime, the Rabi-like oscillations are damped. The absorption of the ac field by the Kitaev spin model is finite and manifests resonance features. Such a behavior is generic for quantum spin liquids with fermionic excitations, and it is different from the linear spin-wave response of magnetically ordered systems to such a parametric pumping.
Majorana edge states in Kitaev chains possessing an effective time reversal symmetry with one fermionic site per unit cell are studied. For a semi-infinite chain the equations for the wave functions of Majorana zero modes can be reduced to a single Wiener-Hopf equation, which has an exact analytical solution. We use this solution to determine the asymptotic behaviors of the Majorana wave functions at large distances from the edge of the chain for several infinite-range models described in the literature with focus on a model with slow power-law falloff of pairing and hopping amplitudes. For these models we also determine the asymptotic behavior of the energy of the fermionic state composed of two Majorana modes in the limit of long (finite) Kitaev chains.
We consider the Kitaev chain model with finite and infinite range in the hopping and pairing parameters, looking in particular at the appearance of Majorana zero-energy modes and massive edge modes. We study the system both in the presence and in the absence of time-reversal symmetry, by means of topological invariants and exact diagonalization, disclosing very rich phase diagrams. In particular, for extended hopping and pairing terms, we can get as many Majorana modes at each end of the chain as the neighbors involved in the couplings. Finally, we generalize the transfer matrix approach useful to calculate the zero-energy Majorana states at the edges for a generic number of coupled neighbors.
The Kitaev chain has been extensively explored in the context of uniform couplings, with studies focusing either on purely nearest-neighbor interactions or on systems dominated by long-range superconducting pairing. Building on these investigations, we introduce a hybrid Kitaev chain in which the lattice is partitioned into two segments: the left segment comprises nearest-neighbor couplings, while the right segment incorporates long-range pairing. To probe the role of the interface, we study two scenarios: a decoupled (suppressed hopping) case, where the segments are isolated, and a coupled case, where they are connected via interface hopping that enables tunneling. Using this setup, we investigate the behavior of Majorana zero modes at the interface between the two segments, finding that in the decoupled case, Majorana zero modes remain sharply localized at the left segment chain edges while massive Dirac modes remain in right segment chain edges, with their energies and localization strongly dependent on the long-range pairing exponent. Introducing a finite interface coupling enables transfer of Majorana zero modes from the edges of the left segment to those of the right segment of the chain. We characterize this dynamics by the fidelity of state transfer, dynamical rotation, and inverse participation ratio. We show the signature of Majorana zero mode transfer across the interface by the spatiotemporal profile of the probability distribution of the time evolved state.
We investigate Majorana modes in a quantum spin chain with bond-dependent exchange interactions by studying its dynamics. Specifically, we consider two-time correlations for the anisotropic Kitaev-Heisenberg (KH) Hamiltonian close to the so-called Kitaev critical point. Here the model coincides with a phase boundary of two uncoupled instances of Kitaev's model for ppp-wave superconductors, together supporting a degenerate ground state characterized by multiple Majorana modes. In this regime, the real-time dynamics of local spins reveal a set of strong zero modes, corresponding to a set of protruding frequencies in the two-time correlation function. We derive perturbative interactions that map the KH spin chain onto the topological regime of Kitaev's fermionic model, thus opening up a bulk gap while retaining almost degenerate modes in the mesoscopic regime, i.e., for finite system sizes. This showcases the emergence of Majorana modes in a chain of effective dimers. Here, the binding energy within each unit cell competes with the interdimer coupling to generate a finite-size energy gap, in analogy with local energy terms in the transverse-field Ising model. These modes give rise to long coherence times of local spins located at the system edges. By breaking the local symmetry in each dimer, one can also observe a second class of Majorana modes in terms of a beating frequency in the two-time correlations function of the edge spin. Furthermore, we develop a scenario for realizing these model predictions in ion-trap quantum simulators with collective addressing of the ions.
The time dependence of the response of the quantum spin liquid with fermionic excitations to parametric pumping has been calculated exactly for the special case of periodic pumping, the steplike periodic field, without using the resonance approximation, and for any value of the pumping magnitude. In the closed regime each mode (related to each eigenstate of the system) oscillates with time. The oscillations persist about the steady-state values, which are determined by the parameters of the pumping. The magnitude of the oscillations is finite for any value of the magnitude of the pumping. Those features drastically differ from the well-known resonance parametric pumping of magnons in magnetically ordered system, where the number of resonance magnons grows exponentially in time. The interference of infinitely many modes generically yields only decaying-in-time modulated oscillations of the total number of quasiparticles per mode even in the closed regime. For the fermionic system only one mode is in resonance, while others are out of resonance. It is totally different from the bosonic behavior of magnons under parametric pumping, where all modes can exist in resonance. The inclusion of the linear relaxation in the open regime for that quantum spin liquid produces the decay of oscillations to the initial state.