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arXiv:2608.19178

Chiral bosonic mean-field Ansatz and spin dynamics in spin-1 Kitaev magnets

Daiki Sasamoto, Arnaud Ralko, Jaime Merino, Joji Nasu

Published Aug 19, 2026• 0 citations
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Abstract

The Kitaev model is a paradigmatic system for realizing quantum spin liquids, but its higher-spin extensions are not exactly solvable, and their spin dynamics is less well understood than in the spin-1/2 case. In this work, we reexamine a previously introduced triplet-pairing $ϕ_t = π/2$ phase pattern for the antiferromagnetic $S = 1$ Kitaev model and extend the analysis to weak symmetric off-diagonal exchanges $Γ$ and $Γ'$. Using a bond-operator formulation of Schwinger-boson mean-field theory, we calculate the dynamical spin structure factor for the triplet 0-flux and triplet $π/2$-flux Ansätze with a spin-correlation scheme appropriate for Kitaev interactions. In the pure Kitaev limit, the $π/2$-flux Ansatz yields a flatter spectrum than the 0-flux Ansatz. The real-space spin correlations show that the $π/2$-flux Ansatz suppresses longer-distance correlations more strongly than the 0-flux Ansatz, yielding a correlation pattern closer to the short-ranged form expected in the Kitaev limit. This comparison shows that the flatness of $S(\boldsymbol{q}, ω)$ is tied to short-ranged spin correlations and is therefore an important consistency check, although it is not, by itself, a diagnostic of time-reversal-symmetry breaking. We then study weak off-diagonal exchanges along $Γ' = Γ$ near the pure Kitaev limit, taking the same-sign relation from analyses of candidate spin-1 Kitaev materials. Gapped solutions are obtained within the constrained $π/2$-flux manifold, and the spectra share the qualitative energy- and momentum-space features found by finite-size exact diagonalization. Taken together, these results support the triplet $π/2$-flux chiral bosonic Ansatz as a useful mean-field description of spin dynamics near the antiferromagnetic $S = 1$ Kitaev limit with weak off-diagonal exchanges.

DOI
10.48550/arxiv.2608.19178

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The Kitaev model is an exactly solvable model that exhibits a quantum spin liquid as its ground state. While this model was originally proposed as an S = 1 / 2 spin model on a honeycomb lattice, extensions to higher-spin systems have recently attracted attention. In contrast to the S = 1 / 2 case, such higher- S models are not exactly solvable and remain poorly understood, particularly for spin excitations at finite temperatures. Here, we focus on the S = 1 Kitaev model, which has been proposed to host bosonic quasiparticles. We investigate this model using Schwinger boson mean-field theory, where bosonic spinons are introduced as fractional quasiparticles by extending bond operators to address anisotropic spin interactions. We determine the mean-field parameters that realize a quantum spin liquid in both ferromagnetic and antiferromagnetic Kitaev models. Based on this mean-field ansatz, we calculate the dynamical and equal-time spin structure factors. We find that, when one uses the conventional decoupling scheme based on Wick decoupling with respect to spinons to calculate spin correlations, the resultant spin structure factors exhibit a momentum dependence that is not consistent with the sign structure expected from the exchange interaction: They possess a strong spectral weight indicating ferromagnetic (antiferromagnetic) correlations in the antiferromagnetic (ferromagnetic) Kitaev model. To resolve this issue, we propose an alternative scheme for evaluating the spin correlations, which is based on decoupling with respect to the bond operators. We demonstrate that, in our scheme, this discrepancy is removed, and the momentum dependence of the spin structure factors is consistent with the sign of the exchange constant. We also calculate the temperature evolution of the dynamical spin structure factor and find that the continuum observed at zero temperature splits into two distinct structures as the temperature increases, which can be understood in terms of the bandwidth narrowing of spinons. Finally, we clarify the origin of why the two distinct decoupling schemes result in different momentum dependences of the spin structure factors and discuss their relationship to results obtained in previous studies.
Motivated by the possibility of finding a bosonic quantum spin liquid in the integer spin-SSS Heisenberg-Kitaev model on the honeycomb lattice, we derive a Schwinger boson mean-field theory involving both singlet and triplet pairing channels which includes hopping and pairing operators on equal footing. The mixed construction introduced here is justified by the good comparison with exact diagonalization energies of the $S\ensuremath{\le}3/2$ Heisenberg-Kitaev model and the perfect match with the Luttinger-Tisza semiclassical energies obtained at large SSS. We find various competing gapped quantum spin liquids close to the Kitaev point. A comparison of their spin excitation spectrum with the dynamical structure factor obtained from exact diagonalizations allows us to identify the physical spin liquid Ansatz of the model. In particular, we identify a chiral quantum spin liquid state whose spin excitation spectrum follows closely the exact diagonalization data and survives up to large spin $S\ensuremath{\lesssim}2$. We propose this state as a promising quantum spin liquid candidate for the integer spin-SSS antiferromagnetic Kitaev model which may be realized in S=1S=1S=1 Kitaev materials A3Ni2XO6{\mathrm{A}}_{3}{\mathrm{Ni}}_{2}{\mathrm{XO}}_{6}A3​Ni2​XO6​ and KNiAsO4{\mathrm{KNiAsO}}_{4}KNiAsO4​.
The star lattice hosts the Yao-Kivelson chiral spin liquid, an exactly solvable Kitaev spin liquid that breaks time reversal spontaneously and is non-Abelian or Abelian depending on the ratio of its two couplings. A density-matrix renormalization group study found that an added Heisenberg exchange leaves it a wide stability window for ferromagnetic but only a narrow one for antiferromagnetic Kitaev coupling. Using projective symmetry groups (PSGs) of fermionic partons, we show that the chirality is not a peculiarity of the solvable point: whenever the πππ spin rotations act on the partons as in the exact solution (spin-gauge locking), a time-reversal-symmetric mean-field state has no amplitude on the triangles, at any range, so a Kitaev-type spin liquid on this lattice must break time reversal. The Abelian and non-Abelian liquids of both signs share a single chiral PSG, within which their transition is a band inversion. The Klein duality, which reverses the Heisenberg coupling, permutes the PSGs and leaves exactly the locked ones unchanged; a Z2Z_2Z2​ spin-liquid phase mapped onto itself is therefore locked and, if it spans a connected range of couplings, contains a Kitaev point. Unprojected mean-field theory in Kitaev's Majorana representation treats both signs alike. Exact second-order perturbation theory about the Kitaev points locates the asymmetry: the triangles respond identically for both signs, whereas on the bonds between triangles the two off-diagonal Heisenberg terms create the same four visons and interfere constructively for antiferromagnetic and almost destructively for ferromagnetic coupling, admixing a low-energy four-vison state 828282 times more strongly on the antiferromagnetic side. The PSG results rest on a classification of 352352352 Z2Z_2Z2​ PSGs with spin-orbit-coupled amplitudes, which also identifies the fermionic partners of the known Schwinger-boson spin liquids.
We calculate dynamical spin structure factors for gapped chiral spin liquid states in the spin-1/2 Heisenberg antiferromagnet on the kagome lattice using Schwinger-boson mean-field theory. In contrast to static (equal-time) structure factors, the dynamical structure factor shows clear signatures of time-reversal symmetry breaking for chiral spin liquid states. In particular, momentum inversion $\mathbit{k}\ensuremath{\rightarrow}\ensuremath{-}\mathbit{k}$ symmetry as well as the sixfold rotation symmetry around the $\mathrm{\ensuremath{\Gamma}}$ point are lost. We highlight other interesting features, such as a relatively flat onset of the two-spinon continuum for the cuboc1 state. Our work is based on the projective symmetry group classification of time-reversal symmetry breaking Schwinger-boson mean-field states by Messio, Lhuillier, and Misguich.
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The S=1/2S=1/2S=1/2 Kitaev model provides an exactly solvable setting in which spin fractionalization produces itinerant Majorana fermions coupled to an emergent Z2\mathbb{Z}_2Z2​ gauge field. In realistic Kitaev magnets, however, non-Kitaev interactions break integrability and make their finite-temperature dynamics difficult to calculate. We develop a finite-temperature parton mean-field theory based on the Abrikosov-fermion representation and incorporate the effects of exchange interactions through the random phase approximation to calculate the dynamical spin structure factor. For the pure Kitaev model, the method reproduces a strong, nearly dispersionless peak at a finite low energy together with a broad continuum at higher energies. It also reproduces the momentum dependence of the dynamical spin structure factor and its temperature evolution, including the downward shift of the low-energy spectral weight, in qualitative agreement with continuous-time quantum Monte Carlo calculations. Within the present theory, the contraction of the continuum is traced to the temperature-induced reduction of the bond amplitudes, which narrows the parton bands and compresses the two-parton phase space. For ferromagnetic Kitaev coupling, the energy of the low-energy peak is minimized at the ΓΓΓ point for a ferromagnetic Heisenberg interaction and maximized there for an antiferromagnetic interaction. In the ferromagnetic Kitaev-ΓΓΓ model, the static response develops a finite-momentum softening, indicating an instability toward the period-three magnetic order reported in numerical studies. For uniform-ferromagnetic states selected by a [111][111][111] magnetic field, we find that a broad parton continuum coexists with magnonlike dispersive structures, producing spectral features consistent with a spinon-to-magnon crossover.