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arXiv:2105.14562 · Physical review. B./Physical review. B

Vortex creation and control in the Kitaev spin liquid by local bond modulations

Seong‐Hoon Jang, Yasuyuki Kato, Yukitoshi Motome

Published Aug 23, 2021• 18 citations
arXivHTMLPDF

Abstract

The Kitaev model realizes a quantum spin liquid where the spin excitations are fractionalized into itinerant Majorana fermions and localized ${\mathbb{Z}}_{2}$ vortices. Quantum entanglement between the fractional excitations can be utilized for decoherence-free topological quantum computation. Of particular interest is the anyonic statistics realized by braiding the vortex excitations under a magnetic field. Despite the promising potential, the practical methodology for creation and control of the vortex excitations remains elusive thus far. Here we theoretically propose how one can create and move the vortices in the Kitaev spin liquid. We find that the vortices are induced by a local modulation of the exchange interaction; especially, the local Dzyaloshinskii-Moriya (symmetric off-diagonal) interaction can create vortices most efficiently in the (anti)ferromagnetic Kitaev model, as it effectively flips the sign of the Kitaev interaction. We test this idea by performing the ab initio calculation for the candidate material $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{Ru}{\mathrm{Cl}}_{3}$ through the manipulation of the ligand positions that breaks the inversion symmetry and induces the local Dzyaloshinskii-Moriya interaction. We also demonstrate a braiding of vortices by adiabatically and successively changing the local bond modulations.

DOI
10.1103/physrevb.104.085142

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The Kitaev model realizes a quantum spin liquid where the spin excitations are fractionalized into itinerant Majorana fermions and localized Z2{\mathbb{Z}}_{2}Z2​ vortices. Quantum entanglement between the fractional excitations can be utilized for decoherence-free topological quantum computation. Of particular interest is the anyonic statistics realized by braiding the vortex excitations under a magnetic field. Despite the promising potential, the practical methodology for creation and control of the vortex excitations remains elusive thus far. Here we theoretically propose how one can create and move the vortices in the Kitaev spin liquid. We find that the vortices are induced by a local modulation of the exchange interaction; especially, the local Dzyaloshinskii-Moriya (symmetric off-diagonal) interaction can create vortices most efficiently in the (anti)ferromagnetic Kitaev model, as it effectively flips the sign of the Kitaev interaction. We test this idea by performing the ab initio calculation for the candidate material \ensuremathα\ensuremath-RuCl3\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{Ru}{\mathrm{Cl}}_{3}\ensuremathα\ensuremath-RuCl3​ through the manipulation of the ligand positions that breaks the inversion symmetry and induces the local Dzyaloshinskii-Moriya interaction. We also demonstrate a braiding of vortices by adiabatically and successively changing the local bond modulations.
The Kitaev quantum spin liquid possesses two fractional quasiparticles, itinerant Majorana fermions and localized visons. It provides a promising platform for realizing a Majorana zero mode trapped by a vison excitation. This local mode behaves as a non-Abelian anyon capable of applications to quantum computation. However, creating, observing, and manipulating visons remain challenging even in the pristine Kitaev model. Here, we propose a theory to control visons enabled by a time-dependent local magnetic field in the Kitaev spin liquid. Examining the time evolution of the magnetic state, we demonstrate that a vison follows a locally applied field sweeping in the system. We clarify that one can move a vison accompanied by a Majorana zero mode by choosing the velocity and shape of the local field appropriately. In particular, the controllability of visons using local fields shows nonlinear behavior for its strength, which originates from interactions between Majorana fermions and visons. The present results suggest that itinerant Majorana fermions other than zero modes play a crucial role in vison transport. Our finding will offer a guideline for controlling Majorana zero modes in the Kitaev quantum spin liquid.
Quantum spin liquids realized in the Kitaev model offer a platform for fractionalization of spin into two quasiparticles: itinerant Majoranas and localized visons. Introducing a uniform weak magnetic field associates a Majorana zero mode with each vison excitation. The vison accompanied by a Majorana zero mode is known to behave as a non-Abelian anyon, which has garnered significant attention for its potential applications in topological quantum computing. Although spatial and temporal control of these anyons is essential for exploring their applicability in quantum computing, numerical simulations of creating, moving, and annihilating non-Abelian anyons by an external field remain challenging as this field violates the exact solvability of the Kitaev model. Moreover, such a field to control non-Abelian anyons may disturb the quantum state due to the energy injection it causes. In this study, by introducing energy dissipation phenomenologically in real-time simulations, we demonstrate that the generation, movement, and annihilation of vison excitations can be achieved while maintaining their localization. In particular, we find that a vison can be moved in a desired direction by using time-dependent local magnetic fields or gradient magnetic fields, and it remains accompanied by a Majorana zero mode even after its movement. We also reveal that a larger spatial extent of the Majorana zero modes bound to a vison facilitates the movement of the vison with smaller field gradients. Furthermore, our numerical simulations demonstrate pair creation and annihilation of visons, triggered by time-dependent magnetic fields. The results obtained in this study highlight the significance of energy dissipation in controlling non-Abelian anyons, which will stimulate further investigations into the nonequilibrium dynamics of fractional quasiparticles in strongly correlated electron systems, as well as studies for applications in topological quantum computation.
We study the possibility to realize a Majorana zero mode that is robust and may be easily manipulated for braiding in quantum computing in the ground state of the Kitaev model in this work. To achieve this we first apply a uniform [111] magnetic field to the gapless Kitaev model and turn the Kitaev model to an effective p + ip topological superconductor of spinons. We then study possible vortex binding in such system to a topologically trivial spot in the ground state. We consider two cases in the system: one is a vacancy and the other is a fully polarized spin. We show that in both cases, the system binds a vortex with the defect and a robust Majorana zero mode in the ground state at a weak uniform [111] magnetic field. The distribution and asymptotic behavior of these Majorana zero modes are studied. The Majorana zero modes in both cases decay exponentially in space, and are robust against local perturbations and other Majorana zero modes far away, which makes them promising candidates for braiding in topological quantum computing.
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.
Non-Abelian anyonic excitations of quantum spin liquids have potential for application to topological quantum computation, but designing logical operations requires developing protocols to faithfully create, move, and read out such quasiparticles. In this paper, we test two such protocols for manipulating Z 2 fluxes (“visons”) of ferromagnetic and antiferromagnetic Kitaev models perturbed by a small uniform magnetic field. We show that a protocol that employs a driving magnetic field locally applied to the sites of the lattice suffers from significant unavoidable errors. These errors, accumulating over time, prevent reaching high probabilities of generating and displacing flux pairs. However, individual spin couplings can also be turned on and off, thus it is possible to create and displace fluxes with nearly 100 % probability. We show that braiding non-Abelian anyons can be achieved with good precision with our second protocol. We comment on the feasibility of our protocols in solid-state devices and quantum emulators.
We investigate how the Kitaev spin liquid state described by Majorana fermions coupled with Z2{Z}_{2}Z2​ gauge fields is affected by non-Kitaev interactions which exist in real candidate materials of the Kitaev magnet. It is found that the off-diagonal exchange interaction referred to as the ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ term dramatically changes the Majorana band structure in the case of the antiferromagnetic Kitaev interaction and gives rise to a topological phase transition from a non-Abelian topological phase with the Chern number equal to $\ifmmode\pm\else\textpm\fi{}1$ to an Abelian phase with the Chern number equal to $\ifmmode\pm\else\textpm\fi{}2$, in which the Z2{Z}_{2}Z2​ vortices behave as Abelian anyons. On the other hand, other non-Kitaev interactions such as the Heisenberg interaction and the $\mathrm{\ensuremath{\Gamma}}$ term only affect the bandwidth of the Majorana band as long as the spin liquid state is not destabilized.