Abstract We formulate a Majorana mean-field theory for the extended J-K-Gamma Kitaev model in a magnetic Zeeman field of arbitrary direction, and apply it for studying spatially inhomogeneous states harboring vortices. This mean-field theory is exact in the pure Kitaev limit and captures the essential physics throughout the Kitaev spin liquid phase. We determine the charge profile around vortices and the corresponding quadrupole tensor. The quadrupole-quadrupole interaction between distant vortices is shown to be either repulsive or attractive, depending on parameters. We predict that electrically biased scanning probe tips enable the creation of vortices at preselected positions. Our results open new perspectives for the electric manipulation of Ising anyons in Kitaev spin liquids.
Abstract We formulate a Majorana mean-field theory for the extended J-K-Gamma Kitaev model in a magnetic Zeeman field of arbitrary direction, and apply it for studying spatially inhomogeneous states harboring vortices. This mean-field theory is exact in the pure Kitaev limit and captures the essential physics throughout the Kitaev spin liquid phase. We determine the charge profile around vortices and the corresponding quadrupole tensor. The quadrupole-quadrupole interaction between distant vortices is shown to be either repulsive or attractive, depending on parameters. We predict that electrically biased scanning probe tips enable the creation of vortices at preselected positions. Our results open new perspectives for the electric manipulation of Ising anyons in Kitaev spin liquids.
The Kitaev honeycomb model realizes a quantum spin liquid where spin degrees of freedom are fractionalized into itinerant Majorana fermions and localized vortices. The entanglement among these particles holds promise for executing fault-tolerant topological quantum computation. While several materials have been proposed to host a quantum spin liquid phase, concrete evidence confirming its existence remains elusive. To aid in identifying this phase, we propose an electrical test using the charge profile, offering a signature for the Z2 vortices in Kitaev materials. This study explores the electric charge response in Kitaev materials, considering a generalized spin model including Heisenberg, off-diagonal exchange interactions, and magnetic Zeeman field. Majorana mean-field theory is employed to analyze spin correlations around vortices, revealing spatial anisotropy in the charge distribution, quantified by the electric quadrupole moment. This mean-field approach remains exact in the pure Kitaev limit and captures the fundamental physics across the Kitaev spin liquid phase. The quadrupole-quadrupole interaction between distant vortices is shown to be either repulsive or attractive, depending on the parameters. We predict that electrically biased scanning probe tips can help create vortices at predetermined locations. The findings of our study pave the way for electrically manipulating Ising anyons in Kitaev spin liquids, offering a promising path for technological applications of Kitaev materials.
We show that spin-spin correlations in a non-Abelian Kitaev spin liquid are associated with a characteristic inhomogeneous charge density distribution in the vicinity of Z_{2} vortices. This density profile and the corresponding local electric fields are observable, e.g., by means of surface probe techniques. Conversely, by applying bias voltages to several probe tips, one can stabilize Ising anyons (Z_{2} vortices harboring a Majorana zero mode) at designated positions, where we predict a clear Majorana signature in energy absorption spectroscopy.
Magnetic fields can give rise to a plethora of phenomena in Kitaev spin systems, such as the formation of nontrivial spin liquids in two and three spatial dimensions. For the original honeycomb Kitaev model, it has recently been observed that the sign of the bond-directional exchange is of crucial relevance for the field-induced physics, with antiferromagnetic couplings giving rise to an intermediate spin liquid regime between the low-field gapped Kitaev spin liquid and the high-field polarized state, which is not present in the ferromagnetically coupled model. Here, by employing a Majorana mean-field approach for a magnetic field pointing along the [001] direction, we present a systematic study of field-induced spin liquid phases for a variety of two and three-dimensional lattice geometries. We find that antiferromagnetic couplings generically lead to (i) spin liquid phases that are considerably more stable in field than those for ferromagnetic couplings, and (ii) an intermediate spin liquid phase which arises from a change in the topology of the Majorana band structure. Close inspection of the mean-field parameters reveal that the intermediate phase occurs due to a field-driven sign change in an `effective' z-bond energy parameter. Our results clearly demonstrate the richness of the Majorana physics of the antiferromagnetic Kitaev models, in comparison to their ferromagnetic counterparts.
The Kitaev model is a unique exactly solvable two-dimensional quantum spin system that exhibits a spin-liquid ground state. Here, the authors rigorously rederive and extend a Majorana mean-field theory approach that reproduces, but applies beyond, the exact results. The authors use this approach to predict inelastic neutron scattering (INS) results on realistic Hamiltonians of hypothetical Kitaev spin-liquid materials. Not limited to just the Kitaev model, this approach can extend the applicability of mean-field theory in analyzing dynamics for any system without a large coordination number.
We investigate the effect of a magnetic field on the extended Kitaev spin-liquid state through phonon dynamics. Using a constrained fermionic self-consistent mean field method, we analyze the quantum spin liquid (QSL) ground state for the extended Kitaev model with both the Zeeman term and the perturbative three-spin interaction term $\ensuremath{\kappa}$. Our results demonstrate the dependence of the stability of the Kitaev QSL state on the field direction, consistent with findings in the literature. Additionally, we calculate the phonon dynamics for acoustic phonons coupled to the Majorana fermion excitations of the Kitaev spin-liquid state, discussing the temperature and field evolution of these quantities.
The Kitaev model is an exactly solvable quantum spin model within the language of constrained real fermions. In spite of numerous studies for magnetic fields along special orientations, there is a limited amount of knowledge on the complete field-angle characterization, which can provide valuable information on the existence of fractionalized excitations. For this purpose, we first study the pure ferromagnetic and antiferromagnetic Kitaev models for arbitrary external magnetic field directions via a mean-field theory, showing that there are many topological phases with different (or zero) Chern numbers, depending on the magnetic field strength and orientations. However, a realistic description of the candidate Kitaev materials, within the edge-sharing octahedra paradigm, requires additional coupling terms, including a large off-diagonal term $\mathrm{\ensuremath{\Gamma}}$ along with possible anisotropic corrections ${\mathrm{\ensuremath{\Gamma}}}_{p}$. It is therefore not sufficient to rely on the topological properties of the bare Kitaev model as the basis for the observed thermal Hall-conductivity signals, and an understanding of these extended Kitaev models with a complete field response is demanded. Starting from the zero-field phase diagram of $\mathrm{K}\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}{\mathrm{\ensuremath{\Gamma}}}_{p}$ models, we identify, besides the Kitaev spin liquid phase, antiferromagnetic zigzag, ferromagnetic phases, as well as an unusual Kitaev(-$\mathrm{\ensuremath{\Gamma}}$) spin liquid phase. The magnetic field response of these phases for arbitrary field orientations provides a remarkably rich phase diagram. For an extended parameter range and just above the critical field where the zigzag phase is suppressed, there is an intermediate phase region with suppressed energy gaps and substantial spin fractionalization. To comply our findings with experiments, we also reproduce a large asymmetry in the extent of this intermediate phases specifically for the two different field directions $\ensuremath{\theta}=\ifmmode\pm\else\textpm\fi{}{60}^{o}$ with respect to the normal to the plane of the honeycomb lattice.