Schwinger boson theory for $S=1$ Kitaev quantum spin liquids

TL;DR

This work develops a Schwinger-boson mean-field theory for the $S=1$ Kitaev model on the honeycomb lattice by extending bond operators to include $$SU(2)$$-breaking channels, enabling a self-consistent description of a quantum spin-liquid ground state with a small spinon gap. It shows that a bond-operator-based decoupling scheme (decoupling II) yields physically sensible spin structure factors and resolves unphysical momentum features seen with conventional Wick decoupling, while finite-temperature dynamics reveal a two-structure continuum from spinon bandwidth narrowing. The results illuminate the finite-temperature spin dynamics of higher-spin Kitaev systems and clarify methodological differences with prior studies, offering a framework to relate bosonic spinons to Majorana-parton pictures and to study experimental signatures in candidate Kitaev materials.

Abstract

The Kitaev model is an exactly solvable model with a quantum spin liquid 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 is proposed to host bosonic quasiparticles. We investigate this model using Schwinger boson mean-field theory, introducing bosonic spinons 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 ansatz, we calculate dynamical and equal-time spin structure factors. We find that the conventional scheme based on Wick decoupling with respect to spinons yields unphysical momentum dependence: it produces strong spectral weight indicating ferromagnetic (antiferromagnetic) correlations in the antiferromagnetic (ferromagnetic) Kitaev model. To resolve this issue, we propose an alternative evaluation based on decoupling with respect to bond operators. We demonstrate that, in our scheme, such unphysical behavior disappears and the momentum dependence of the spin structure factors is consistent with the sign of the exchange constant. We also compute the temperature evolution of the dynamical spin structure factor and find that the zero-temperature continuum splits into two distinct structures as temperature increases, which can be understood in terms of the bandwidth narrowing of spinons. Finally, we clarify why the two decoupling schemes result in different momentum dependences and discuss their relationship to previous studies.

Figures (8)

• Figure 1: (a) Schematic picture of the honeycomb lattice on which the $S=1$ Kitaev model is defined. The blue, green, and red lines denote the $x$, $y$, and $z$ bonds, respectively. The gray arrows indicate the primitive translation vectors $\bm{a}_{1}$ and $\bm{a}_{2}$ of the four-sublattice unit cell, which are used in the present calculations. Within the four-sublattice unit cell, the independent mean-field parameters are explicitly distinguished on the bonds by differences in color and line style. Arrows on the bonds indicate the directions of the mean fields $\left(\braket{\mathcal{A}_{ij}},\braket{\mathcal{B}_{ij}},\braket{\mathcal{C}^{\gamma}_{ij}},\braket{\mathcal{D}^{\gamma}_{ij}}\right)$ from site $i$ to $j$. (b) First and extended Brillouin zones of the original honeycomb lattice. Filled symbols denote the high-symmetry points. The gray arrows indicate the primitive reciprocal lattice vectors $\bm{b}_{1}$ and $\bm{b}_{2}$ corresponding to $\bm{a}_{1}$ and $\bm{a}_{2}$ in the real space.
• Figure 2: Mean-field pattern assumed in the present SBMFT for the $S=1$ antiferromagnetic Kitaev model. Among the mean-field parameters, only $\braket{\mathcal{D}_{ij}^{\gamma}}$ is nonzero on each $\gamma$ bond and takes a common value, reflecting the lattice $C_{3}$ symmetry. Since $\mathcal{D}_{ij}^{\gamma}=\mathcal{D}_{ji}^{\gamma}$, the fields are independent of bond orientation; accordingly, the arrows used in Fig. \ref{['fig:mean_field_ansatz']}(a) are omitted here.
• Figure 3: Dynamical (upper row) and equal-time (lower row) spin structure factors in the ground state of the $S=1$ Kitaev model. In (a)--(c), the spectra are shown along the path connecting the high-symmetry points indicated in Fig. \ref{['fig:mean_field_ansatz']}. In (d)--(f), the inner and outer hexagons represent the first and extended Brillouin zones, respectively. Panels (a) and (d) correspond to results obtained using decoupling I in the AFM model ($K>0$), panels (b) and (e) to results obtained using decoupling II in the AFM model, and panels (c) and (f) to results obtained using decoupling II in the FM model ($K<0$).
• Figure 4: Finite-temperature dynamical spin structure factor $S(\bm{q},\omega)$ in the FM and AFM Kitaev models at (a),(d) $T=0.1$, (b),(e) $T=0.2$, and (c),(f) $T=0.3$. Panels (a)--(c) are the results in the AFM model ($K>0$), whereas panels (d)--(f) are for the FM model ($K<0$).
• Figure 5: (a) Spinon dispersions in the $S=1$ AFM Kitaev model along the path connecting the high-symmetry points indicated in Fig. \ref{['fig:mean_field_ansatz']}. At $T=0$, the spectrum is nearly gapless, but a small gap remains. (b) Zero-temperature excitation gap $\omega_{\text{min}}/|K|$ of spinons as a function of the spin quantum number $S$.