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Physical review. B./Physical review. B

Parametric pumping of the two-dimensional quantum spin liquid

Andrei A. Zvyagin

Published Feb 24, 2017• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.95.064428

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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.
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.
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.
The response of the one-dimensional antiferromagnetic spin-1/2 chain to the ac staggered magnetic field is studied within the Hartree-Fock-like approximation. For a closed system that field produces oscillations of the magnetization with the frequency of the ac field, modulated by the large number of small Rabi frequencies. In the open system the Rabi oscillations decay due to relaxation processes. It is shown that those oscillations also contribute to the dynamics of the entanglement of the system. The calculated temperature and frequency dependence of the dynamical characteristics is strictly nonmonotonic, reflecting the complicated character of the common action of the homogeneous and staggered dc field.
The Kekul\'e-Kitaev model, an extension of the Kitaev model, exhibits quantum spin liquid properties, which has an exact solution through Kitaev parton construction. In this study, we calculate the dynamical spin structure factor as a linear response and the third-order magnetic susceptibility as a nonlinear response using two-dimensional coherent spectroscopy for the Kekul\'e-Kitaev model. Our results reveal that the few-matter fermion excitations approximation provides reliable results for both linear and nonlinear responses. Notably, while the Kekul\'e-Kitaev model shows linear and nonlinear responses qualitatively similar to the Kitaev model, it displays distinct behavior under a weak uniform and staggered magnetic field. Specifically, the Kekul\'e-Kitaev model does not present a non-Abelian phase under a uniform magnetic field, while such phase appears in the presence of a staggered magnetic field. Interestingly, within the non-Abelian phase, signals originating from two nonadjacent fluxes in the nonlinear response are stronger than signals from other flux excitations. Furthermore, we demonstrate that the ground state of the Kekul\'e-Kitaev model at isotropic coupling is mapped, through unitary spin rotations, to an excited state of the Kitaev model with a uniform flux configuration.
We study the response of a Kitaev spin liquid (KSL) to a local magnetic field perpendicular to the Kitaev honeycomb lattice. The local magnetic field induces a dynamical excitation of a flux pair in the spin liquid and the system can be described by a generally particle-hole asymmetric interacting resonant level model. The dynamical excitation of the flux pair closes the flux gap in the spectrum of the spin correlation function locally for the gapless KSL even from the perturbative response to a weak magnetic field. Beyond the perturbative regime, the p-h asymmetry competes with the magnetic field and results in a rich phase diagram. Moreover, the magnetic field breaks the gauge equivalence of the ferromagnetic and antiferromagnetic Kitaev couplings of the ground state and leads to very different behaviors for the two cases. The antiferromagnetic case experiences a first-order phase transition to the polarized state during magnetization, whereas the ferromagnetic case does not. This study can be generalized to the Kitaev model in a uniform magnetic field and may help understand issues in recent experiments on KSL candidates.
The Kitaev model offers a platform for quantum spin liquids (QSLs) with fractional excitations, itinerant Majorana fermions and localized fluxes. Since these fractional excitations could be utilized for quantum computing, how to create, observe, and control them through the spin degree of freedom is a central issue. Here, we study dynamical spin transport in a wide range of frequencies for the Kitaev-Heisenberg model, by applying an AC magnetic field to an edge of the system. We find that, in the Kitaev QSL phase, spin polarizations at the other edge are resonantly induced in a specific spin component, even though the static spin correlations are vanishingly small. This interedge spin resonance appears around the input frequency over the broad frequency range. Comparing with the dynamical spin correlations, we clarify that the resonance is governed by the itinerant Majorana fermions with a broad continuum excitation spectrum, which can propagate over long distances, although it vanishes for the pure Kitaev model because of accidental degeneracy and requires weak Heisenberg interactions. We also find that the spin polarizations in the other spin components are weakly induced at an almost constant frequency close to the excitation gap of the localized fluxes, irrespective of the input frequency. These results demonstrate that the dynamical spin transport is a powerful probe of the fractional excitations in the Kitaev QSL. Possible experimental realization of the interedge spin resonance is discussed.