Proximate quantum spin liquids and Majorana continua in magnetically ordered Kitaev magnets

Abstract

We study the spin excitation spectra in magnetically ordered phases proximate to the Kitaev quantum spin liquid (KQSL). Although the low-energy universal features should be governed by the magnetic orders, the $\textit{non-universal}$ high-energy features of the KQSL and adjacent phases can be remarkably similar. Therefore, we study the extended Kitaev model within a Stoner-like theory using Majorana partons, and compute the inelastic neutron scattering (INS) intensities in the random phase approximation. First, we benchmark against the antiferromagnetic (AFM) Heisenberg model and recover the AFM order with linear Goldstone modes. We then explore the phase diagram which agrees qualitatively with previous numerical results. In particular, the Majorana parton theory accurately captures Order-by-Disorder effects in the Kitaev-Heisenberg limit. We also find large INS intensities near the associated high-symmetry Brillouin zone (BZ) points of the magnetic orders. At intermediate and high energies, broad multi-spinon continua emerge across the BZ, providing a distinct mechanism for magnon decay and spectral broadening beyond the conventional multi-magnon decay scenario. Finally, we study the model Hamiltonian of candidate Kitaev material $α$-RuCl$_3$. The zigzag ground state agrees qualitatively with experiments, its stability under external magnetic field also exhibits strong anisotropy in the field directions, and broad scattering continua are recovered similar to those observed experimentally.

Figures (6)

• Figure 1: Phase diagram and INS intensities of the proximate KQSL in the $\mathbf{KJ\Gamma}$-model.a (upper panel) near the FM KQSL phase; b (lower panel) near the AFM KQSL phase. The phase diagrams are adapted schematically from Ref. rau2014generic. The momentum path convention is shown in Fig. \ref{['main_v2_1:fig:unitcell']}. The INS intensity is plotted in logarithmic scale introduced in Sec. \ref{['main_v2_1:sec:spin-susceptibility']}.
• Figure 2: Lattice and Brillouin zone (BZ) conventions. The unit cells are shown in black dashed lines with: a two sublattices (KQSL, FM, AFM); b four sublattices (zigzag, stripy); c six sublattices ($120^\circ$). The first and second BZs of the honeycomb lattice are shown in gray and black solid lines respectively. The BZs of the four- and six-sublattice unit cells are shown in grey dashed lines.
• Figure 3: INS intensities for the Kitaev-Heisenberg model inside the AFM phase at $J>0$. The intensities are shown in logarithmic scale introduced in Sec. \ref{['main_v2_1:sec:spin-susceptibility']} which makes the magnon mode more visible. In a, c-d the off-set value $c=1$ introduced below Eq. \ref{['main_v2_1:eq:INS-intensity']}. In b we set $c=10^{-4}$ to highlight the low-intensity features, compared to a which focuses on the sharp gapless Goldstone mode. The apparent anisotropy of the Goldstone mode spin velocity in a-b is due to that the momentum path lengths $M_1\rightarrow K_1$, $K_1\rightarrow K'$ are scaled differently.
• Figure 4: INS intensity for $K<0$ and $J = -0.11 |K|$ at external magnetic field.a$B=0$ and b$B= 0.157 |K|, \ \mathbf{B}\parallel (111)$. It was shown in Ref. rao2025dynamical that for the field value at b the KQSL is stabilized.
• Figure 5: INS intensity near the transition points for the Kitaev-Heisenberg model with FM $K$. Due to proximity to the critical $J_{\text{cr}}$, both phases admit MF solutions. a in the stripy phase and b the QSL phase at $J = 0.103|K|$. c in the FM phase and d the QSL phase at $J = -0.127|K|$. The KQSL phase intensity is normalized to be the same as that in the corresponding ordered phase.