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.

• The paper introduces a Majorana parton framework combined with RPA to bridge conventional magnon theories and fractionalized Majorana excitations in Kitaev magnets.
• Key results reveal that magnon decay via multi-Majorana continua aligns with experimental inelastic neutron scattering data in materials like α-RuCl₃.
• The study validates a unified approach to capture order-by-disorder effects, anisotropic critical fields, and dynamic responses in proximate quantum spin liquid phases.

Proximate Quantum Spin Liquids and Majorana Continua in Magnetically Ordered Kitaev Magnets

Introduction and Motivation

This work addresses the complex interplay between conventional magnetic order and fractionalized excitations in quantum magnets proximate to Kitaev quantum spin liquid (KQSL) phases. The focus is on theoretically modeling and interpreting the dynamical responses, particularly inelastic neutron scattering (INS) spectra, of extended Kitaev systems where the physics of proximate spin liquids and Majorana fermion continua remains influential even within magnetically ordered regimes. The paper leverages a Majorana parton mean-field theory (MFT) combined with a random phase approximation (RPA) framework to bridge the gap between spin-wave (magnon) theories and the partonic description relevant near the KQSL boundary.

Theoretical Framework and Methodology

The analysis is rooted in the extended Kitaev model ($KJ\Gamma\Gamma'J_3$) on the honeycomb lattice, relevant for materials such as $\alpha$-RuCl$_3$. The Majorana fermion representation is adopted for spins, incorporating both the KQSL regime and adjacent magnetically ordered phases. The mean-field solution is constructed self-consistently to allow for magnetic order (finite magnetization $\mathbf{m}$), and dynamical spin susceptibilities are computed via the RPA.

The theoretical protocol is as follows:

• Spins are mapped onto Majorana fermions with a quartic on-site constraint.
• Mean-field decoupling provides the Majorana band structure and magnetic order parameters.
• The dynamical spin susceptibility is calculated at the one-loop level and dressed via RPA to include interaction effects.
• The approach is benchmarked against the Heisenberg limit and then systematically applied to the $KJ\Gamma$ and $KJ\Gamma\Gamma'J_3$ parameter regimes, including finite external fields.

Key Results and Numerical Findings

Benchmarking and Order-by-Disorder Effects

The parton RPA method recovers the anticipated results in the Heisenberg antiferromagnetic limit, including linear Goldstone modes and correct ordering patterns. Crucially, within the Kitaev-Heisenberg model, the Majorana mean field automatically captures order-by-disorder (ObD) effects: both the selection of spin-quantization axes and the generation of quantum fluctuation-induced energy gaps. The calculated magnon gaps match well with those from non-linear spin-wave theory, especially for antiferromagnetic interactions.

A central claim is that the Majorana mean-field approach incorporates quantum fluctuations at the level required to reproduce these ObD phenomena, in contrast to linear spin-wave theory, which fails without higher-order corrections.

Magnon Decay via Multi-Majorana Continua

The INS spectra in the ordered phases reveal broad, high-energy continua that do not correspond to conventional magnons but to multi-spinon (Majorana) excitations. The paper asserts that this mechanism provides a fundamentally distinct route for magnon damping and decay, going beyond multi-magnon processes central to standard spin-wave theory. The comparison to DMRG and experimental INS data suggests that this broadening is consistent with observations of overdamped modes and continua in $\alpha$-RuCl$_3$.

The approach shows that even on the ordered side of the phase diagram, high-energy features reminiscent of the fractionalized KQSL persist, reflecting a long confinement length and significant proximity effects.

Modelling of Real Materials: $\alpha$-RuCl$_3$

Utilizing a realistic set of coupling parameters for $\alpha$0-RuCl$\alpha$1, the ground state is found to adopt zigzag magnetic order with the orientation of the moment in close agreement with experimental findings (tilt angle $\alpha$2 vs. the experimental $\alpha$3). The computed INS intensity profiles reproduce the broad spectral weight near the $\alpha$4 and $\alpha$5 points and lack of sharp magnon modes, corroborating experimental signatures attributed to underlying fractionalized excitations.

The theory also accounts for the strong anisotropy in critical field values for destroying zigzag order, matching the experimental distinction between in-plane and out-of-plane instability.

Implications and Theoretical Significance

The findings have several significant implications:

• Unified Framework: The Majorana MFT+RPA formalism provides a unified treatment for both sharp magnon modes and broad Majorana continua without ad hoc phenomenological assumptions or parameter fitting, relying solely on microscopic Hamiltonian parameters.
• Parton Descriptions in Ordered Phases: The persistence of Majorana multi-spinon continua in magnetically ordered backgrounds suggests that partonic descriptions retain relevance beyond strict QSL phases, especially in the high-energy regime, challenging the common dichotomy between confined and deconfined regimes.
• Benchmarking Against Experiment: The ability of the formalism to reproduce not only low-energy order but also key features of experimental dynamical response (broadened bands, anisotropic critical fields, continuum scattering) supports its applicability to real materials and the proximate spin liquid paradigm.
• Methodological Insights: The work demonstrates that ObD effects and corresponding pseudo-Goldstone gaps can be partially or fully captured at the mean-field level in a Majorana basis, which does not rely on classical spin Ansatz or higher-order spin-wave corrections.

Limitations and Future Directions

The authors acknowledge that the parton RPA formalism does not fully capture Majorana confinement at low energies in ordered phases, which could lead to sharp, discrete modes at the bottom of the continuum, akin to phenomena established in Ising chain materials. Extending the theory to this intermediate regime poses methodological challenges. Furthermore, the approach can be straightforwardly generalized to provide theoretical predictions for other dynamical response functions, such as Raman and RIXS spectra, and adapted to other lattice models or materials proximate to QSL phases.

Possible theoretical future avenues include:

• Incorporation of gauge fluctuation effects beyond mean-field and RPA.
• Extension to non-equilibrium or finite-temperature dynamical response.
• Application to other fractionalized phases, including U(1) QSLs or topological spin liquids with different gauge structures.
• Cross-correlations with other systems with proximate quantum order, such as the cuprates or materials showing fractional Chern insulator or fractional quantum Hall phases.

Conclusion

This work establishes a versatile and predictive theoretical scheme for exploring dynamical properties of frustrated magnets in the vicinity of Kitaev spin liquids. By combining Majorana mean-field theory with RPA, it quantitatively captures both low-energy magnon phenomena and broad multi-spinon continua, elucidating mechanisms for magnon damping that transcend conventional multi-magnon decay. The theoretical predictions closely align with experimental data in candidate Kitaev magnets such as $\alpha$6-RuCl$\alpha$7, supporting the "proximate spin liquid" interpretation and demonstrating that partonic physics has pronounced effects even in magnetically ordered backgrounds. This framework provides a foundation for further studies of dynamical signatures and fractionalization in a broad class of strongly correlated quantum systems.

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Reference: "Proximate quantum spin liquids and Majorana continua in magnetically ordered Kitaev magnets" (2604.03099)