Abstract
The description of quantized collective excitations stands as a landmark in the quantum theory of condensed matter. A prominent example occurs in conventional magnets, which support bosonic magnons—quantized harmonic fluctuations of the ordered spins. In striking contrast is the recent discovery that strongly spinorbitalcoupled magnets, such as αRuCl_{3}, may display a broad excitation continuum inconsistent with conventional magnons. Due to incomplete knowledge of the underlying interactions unraveling the nature of this continuum remains challenging. The most discussed explanation refers to a coherent continuum of fractional excitations analogous to the celebrated Kitaev spin liquid. Here, we present a more general scenario. We propose that the observed continuum represents incoherent excitations originating from strong magnetic anharmonicity that naturally occurs in such materials. This scenario fully explains the observed inelastic magnetic response of αRuCl_{3} and reveals the presence of nontrivial excitations in such materials extending well beyond the Kitaev state.
Introduction
From magnons in ordered magnets to phonons in periodic crystals, the appearance of bosonic collective excitations is ubiquitous in condensed phases of matter^{1}. For this reason, special attention is given to those states that support more exotic collective modes, for which the conventional paradigm breaks down. In the context of magnetic phases, the breakdown of magnons is commonly thought to require closeness to an unconventional state such as a quantum spin liquid^{2,3,4}. A notable example occurs in Kitaev’s exactly solvable honeycomb model^{5}, for which strongly anisotropic and bonddependent interactions fractionalize conventional spin excitations into Majorana spinons and fluxes. This Kitaev state has recently risen to prominence due to the suggestion that it may be realized in heavy metal 4d ^{5} and 5d ^{5} insulators via a specific interplay between the crystal field and strong spinorbit coupling^{6}, and, consequently, a variety of candidate materials based on Ir^{4+} and Ru^{3+} have been intensively explored^{7}. Encouragingly, evidence of a continuum of magnetic excitations that is inconsistent with conventional magnons was found in the majority of such materials, including the twodimensional (2D) honeycomb Na_{2}IrO_{3} ^{8, 9} and αRuCl_{3} ^{10,11,12,13,14}, as well as the threedimensional (3D) analogs β,γLi_{2}IrO_{3} ^{15}, despite all of them having magnetically ordered ground states.
While the observed excitation continua in these systems have been interpreted in terms of signatures of the Kitaev state, the lowsymmetry crystalline environment of the real materials also allows various additional interactions beyond Kitaev’s model^{16,17,18}, which are thought to be large based on both experimental^{19, 20} and theoretical^{18, 21, 22} considerations. In this sense, understanding the mechanism for the breakdown of magnons and the appearance of a broad continuum of magnetic excitations remain a key challenge.
In this work, we study a representative case αRuCl_{3}, which forms a layered 2D honeycomb lattice and displays zigzag magnetic order below T _{N} ~7 K^{12, 13, 23}. We specifically address the recent inelastic neutron scattering (INS) measurements, which have revealed lowenergy magnons^{24} coexisting with an intense excitation continuum^{12}. The latter continuum possesses a distinctive sixfold star shape in momentum space, and large intensity at the 2D Γpoint over a wide energy range E = 2–15 meV^{12}. To resolve the nature of this continuum, we take two complementary approaches. We first theoretically investigate the neutron spectra over a range of relevant magnetic interactions in order to determine the correct spin Hamiltonian for αRuCl_{3}, which has been a subject of intense recent discussion^{18, 25,26,27,28}. Second, we identify the conditions that lead to the breakdown of conventional magnons in the presence of strongly anisotropic and frustrated interactions, revealing that nontrivial excitations naturally persist well beyond the Kitaev spin liquid.
Results
The model
Based on previous ab initio studies^{18, 25,26,27,28}, the largest terms in the spin Hamiltonian of αRuCl_{3} are generally expected to include nearest neighbor Heisenberg J _{1}, Kitaev K _{1}, and offdiagonal Γ _{1} couplings, supplemented by a third neighbor Heisenberg J _{3} term:
where \(\left\langle {i,j} \right\rangle\) and \(\left\langle {\left\langle {\left\langle {i,j} \right\rangle } \right\rangle } \right\rangle\) refer to summation over first and third neighbor bonds, respectively (see Fig. 1). The bonddependent variables {α, β, γ} distinguish the three types of first neighbor bonds, with {α, β, γ} = {y, z, x}, {z, x, y}, and {x, y, z} for the Xbonds, Ybonds, and Zbonds, respectively. The third neighbor interactions are bondindependent. The phase diagram of this model has been discussed previously^{17, 18, 26, 29}, and is further detailed in Supplementary Note 1; here we review the key aspects.
In the limit J _{1} = Γ _{1} = J _{3} = 0, the ground state is a gapless Z _{2} spin liquid for either positive or negative K _{1}, as demonstrated in Kitaev’s seminal work^{5}. Small perturbations from the pure K _{1} limit may induce various magnetically ordered states, such as the zigzag antiferromagnetic (AFM) state observed in αRuCl_{3} and shown in Fig. 1. The simplest perturbation is the introduction of a finite J _{1}, which yields the wellstudied (J _{1}, K _{1}) nearest neighbor HeisenbergKitaev (nnHK) model. This model hosts zigzag order in the region K _{1} > 0, J _{1} < 0, as discussed in Supplementary Note 1. Accordingly, previous analysis of the powder INS experiments within the context of the nnHK model^{13} suggested that K _{1} ~+7 meV, and \(\left {{J_1}{\rm{/}}{K_1}} \right\sim 0.3  0.7\) for αRuCl_{3}. On this basis, the excitation continua observed experimentally were initially interpreted in terms of proximity to the AFM K _{1} > 0 spin liquid^{12, 13}. However, the further consideration of finite Γ _{1} and J _{3} interactions in Eq. (1) significantly expands the experimentally relevant region, as both interactions generally stabilize zigzag order. Indeed, recent ab initio studies^{18, 25,26,27,28} have suggested that the zigzag order in αRuCl_{3} emerges from \({J_1}\sim 0,\,{K_1} < 0,\,{\mathit \Gamma _1} >0\), and J _{3} > 0, with \(\left {{\mathit \Gamma _1}{\rm{/}}{K_1}} \right\sim 0.5  1.0\) and \(\left {{J_3}{\rm{/}}{K_1}} \right\sim 0.1  0.5\), as reviewed in Supplementary Note 2. That is, K _{1} is ferromagnetic, and supplemented by significant Γ _{1} and J _{3} interactions. Such interactions would represent large deviations from both Kitaev’s original model and the region identified by initial experimental analysis. Before discussing the origin of the excitation continua, it is therefore crucial to first pinpoint the relevant interaction parameters.
In order to address this issue directly, we have computed the neutron scattering intensity \({\cal I}({\bf{k}},\omega )\) for a variety of interactions within the zigzag ordered phase via both linear spinwave theory (LSWT) and exact diagonalization (ED). For the latter case, we combine results from various periodic 20site and 24site clusters compatible with the zigzag state in order to probe a wider range of kpoints (see “Methods” section). Full results for the complete range of models are presented in Supplementary Note 5. Here, we highlight the key results for two representative sets of interactions. Within the (J _{1}, K _{1}) nnHK model, we focus on Model 1 (J _{1} = −2.2, K _{1} = +7.4 meV; \(\left {{J_1}{\rm{/}}{K_1}} \right\) = 0.3), which lies on the border of the region initially identified in ref. ^{12}, close to the spin liquid. Beyond the nnHK model, we consider Model 2 (J _{1} = −0.5, K _{1} = −5.0, Γ _{1} = +2.5, J _{3} = +0.5 meV) for which parameters have been guided by recent ab initio studies^{18, 25,26,27,28}, and further optimized to improve agreement with the experimental spectra. Results for Models 1 and 2 are first presented in Figs. 2 and 3, which show detailed ωdependence and kdependence of \({\cal I}({\bf{k}},\omega )\), along with the evolution of the spectra upon changing parameters toward the K _{1} > 0 or K _{1} < 0 spin liquid regions.
Nearest neighbor HeisenbergKitaev model
We begin by analyzing the spectra \({\cal I}({\bf{k}},\omega )\) within the zigzag phase of the (J _{1}, K _{1}) nnHK model, starting with Model 1 (Fig. 2). Despite proximity to the spin liquid, the ED calculations on Model 1 (Fig. 2b) show sharp dispersive modes appearing over the majority of the Brillouin zone that are consistent with the conventional magnons of LSWT (Fig. 2a). Indeed, for energies below the main spinwave branch (ω = 1.3–2.3 meV), intensity is localized around the Mpoints and Ypoints, corresponding to the pseudoGoldstone modes of the zigzag order (Fig. 2c). ED calculations show clear spinwave cones emerging from such points and extending to higher energies. Large deviations from LSWT are observed only for the highest energy excitations, which appear near the 2D Γpoint for energies ω > 5 meV. Here, the ED calculations display a broad continuumlike feature centered at \(\omega \sim {K_1}\) that resembles the response expected for the K _{1} > 0 Kitaev spin liquid, as shown in Fig. 2d. However, comparison with the experimental \({\cal I}({\rm{\Gamma}} ,\omega )\) shows poor agreement, while the experimental intensity extends from 2 to 15 meV, the ED results for Model 1 predict intensity only at high energies >5 meV. Indeed, the evolution of the Γpoint intensity with \(\left {{J_1}{\rm{/}}{K_1}} \right\) is shown in Fig. 2e. On approaching the K _{1} > 0 spin liquid by decreasing \(\left {{J_1}{\rm{/}}{K_1}} \right\), excitations at the Γpoint shift to higher energy, such that none of the parameters in the vicinity of the spin liquid reproduce the experimental ωdependence of \({\cal I}({\rm{\Gamma}} ,\omega )\). Similar conclusions can also be drawn from recent Density Matrix Renormalization Group (DMRG) studies of the nnHK model^{30}. We therefore conclude that the broad features observed experimentally in \({\cal I}({\rm{\Gamma}} ,\omega )\) at relatively low energies^{12} are incompatible with the nnHK model with J _{1} < 0 and K _{1} > 0.
Extended ab initio guided model
In order to treat the effect of interactions beyond the nnHK model, we consider now the ab initio guided Model 2. In contrast to Model 1, ED calculations on Model 2 (Fig. 3b) show large deviations from standard LSWT (Fig. 3a) over a wide range of k and ω. This model reproduces many of the experimental spectral features^{12, 24}. In particular, sharp singlemagnonlike peaks appear only near the pseudoGoldstone modes at the Mpoints and Ypoints. Elsewhere in the Brillouin zone, broad continuumlike features are observed within the ED resolution. As demonstrated in Fig. 3c, we find significant intensity at low energies (ω < 2.3 meV), at both the Γpoints and (M,Y)points. For the intermediate energy region (ω = 5.5–8.5 meV), \({\cal I}({\bf{k}})\) resembles the sixfold star shape observed in ref. ^{12}. At higher energies (ω > 10.5 meV) scattering intensity is mainly located at the Γpoint, also in accord with experiment. Furthermore, the ED results for the Γpoint intensity \({\cal I}({\rm{\Gamma}} ,\omega )\) show a broad range of excitations peaked around 4 and 6 meV, and extending up to ~15 meV (Fig. 3d). Therefore, ED calculations on Model 2 reproduce all of the main experimental spectral features, validating the range of interactions indicated by ab initio calculations. The only aspect that is not quantitatively reproduced within the Model 2 is the magnitude of the gap at the Mpoint (~0.8 meV at the level of LSWT vs. ~2 meV experimentally^{13, 24}). This discrepancy may result from deviations from C _{3} symmetry, which are allowed within the C2/m space group^{18, 31}, but not considered here for simplicity (see Supplementary Fig. 11). Interestingly, the spectral features at the Γpoint become dramatically sharper on approaching the K _{1} < 0 spin liquid, as shown in the evolution of \({\cal I}({\rm{\Gamma}} ,\omega )\) with the ratio \(\left {{\mathit \Gamma _1}{\rm{/}}{K_1}} \right\) (Fig. 3e). This result reveals that the broad continuum may not be directly associated with a proximity to the Kitaev state.
Magnon stability beyond LSWT
To gain further insight into the reason for such a drastic contrast between the stability of magnons in Models 1 and 2, it is useful to consider possible magnon decay channels in the zigzag ordered phase. At the level of LSWT, the spinwave Hamiltonian is truncated at quadratic order, and can be written \({{\cal H}_2} = \mathop {\sum}\nolimits_{{\bf{k,m}}} {\epsilon _{{\bf{k}},m}}\,a_{{\bf{k}},m}^\dag {a_{{\bf{k}},m}}\) in terms of magnon creation (annihilation) operators a ^{†} (a), where \({\epsilon _{{\bf{k}},m}}\) denotes the dispersion of the mth magnon band. In this harmonic approximation, magnons represent sharp, welldefined excitations. However, when higherorder anharmonic terms are included, the total magnon number \({N_{{\rm{tot}}}} = \mathop {\sum}\nolimits_{{\bf{k}},m} a_{{\bf{k}},m}^\dag {a_{{\bf{k}},m}}\) is typically not a conserved quantity, such that the stability of magnons is not guaranteed beyond quadratic order. Quantum fluctuations associated with the higherorder anharmonic decay terms may mix sharp singlemagnon modes with the multimagnon continuum^{32,33,34}. Similar considerations also apply to the breakdown of other collective modes, such as phonons in anharmonic crystals^{35, 36}. From this perspective, a large decay rate is expected for any singlemagnon mode that is energetically degenerate with the multiparticle continuum, unless there are specific symmetries guaranteeing that the two do not couple. It is therefore useful to consider the prerequisites for magnon breakdown in the presence of the strongly anisotropic interactions of Eq. (1).
Magnon decay channels for the nnHK model
We first examine the stability of magnons in the nnHK model. For pure J _{1} and K _{1} interactions, the total spin projections \(S_{{\rm{tot}}}^\gamma = \mathop {\sum}\nolimits_i S_i^\gamma\) are conserved along the cubic axes γ = {x, y, z} modulo two. Since the ordered moment also lies along one of the cubic axes in the zigzag phase^{20, 37} (see Fig. 4c), the possible magnon decay channels are restricted. In the local picture, the relevant quantum fluctuations are local singlet \(S_i^xS_j^x\left { \uparrow \downarrow } \right\rangle = \left { \downarrow \uparrow } \right\rangle\) and triplet \(S_i^xS_j^x\left { \uparrow \uparrow } \right\rangle = \left { \downarrow \downarrow } \right\rangle\) fluctuations shown in Fig. 4a, with \(\Delta S_{{\rm{tot}}}^z = 0\) and 2, respectively. In the magnon picture, the Hamiltonian can only contain evenorder terms (i.e., \({\cal H} = {{\cal H}_2} + {{\cal H}_4} + ...\)), analogous to conventional Heisenberg antiferromagnets with collinear ordered spins^{32, 34}. For example, the fourthorder decay process due to \({{\cal H}_4}\) mixes the onemagnon states with the threemagnon continuum (ΔN _{tot} = ±2), where
Here, the bold index (n ≡ k _{ n }, m _{ n }) labels both momentum and magnon band. This process is pictured in Fig. 4b. As noted above, the effect of such terms depends crucially on the availability of lowenergy threemagnon states in which to decay.
The density of threemagnon states for Model 1 is shown in Fig. 4d, based on the onemagnon dispersions obtained in LSWT. At each kpoint, the lowest energy threemagnon state \(a_{{{\bf{q}}_1}}^\dag a_{{{\bf{q}}_2}}^\dag a_{{{\bf{q}}_3}}^\dag \left 0 \right\rangle \) (with q _{1} + q _{2} + q _{3} = k) is obtained by placing two particles in the pseudoGoldstone modes at opposite Mpoints (q _{1} + q _{2} = 0), and the third particle at q _{3} = k, with total energy \(E_3^{{\rm{min}}}({\bf{k}}) = {\epsilon _{{\bf{k}},1}} + 2{\epsilon _{{\rm{M}},1}}\). This implies \(E_3^{{\rm{min}}}({\bf{k}}) \ge {\epsilon _{{\bf{k}},1}}\). That is, the threemagnon states lie above the lowest onemagnon band at every kpoint. As a result, every magnon in the lowest band remains kinetically stable, due to the absence of lowenergy threeparticle states in which to decay. Precisely this condition ensures the stability of lowenergy magnons in conventional isotropic antiferromagnets, and explains the sharp magnonlike peaks observed in Fig. 2b for Model 1. Strong spectral broadening in the nnHK model can occur only for highlying excitations with \({\epsilon _{{\bf{k}},m}} >{\epsilon _{{\bf{k}},1}} = E_3^{{\rm{min}}}\), where the density of threemagnon states is finite, such as at the 2D Γpoint. On approaching the spin liquid (at J _{1}/K _{1} = 0), this condition is relaxed due to the vanishing dispersion of the lowest magnon band (i.e., \({\epsilon _{{\bf{k}},1}}\)→0), which corresponds to a vanishing energy cost the singlet fluctuations shown on the left of Fig. 4a. The relevant fluctuations in the limit J _{1}/K _{1}→0 therefore correspond to ΔN _{tot} = ±2. For other values of J _{1}/K _{1}, the majority of magnons are expected to remain stable due to the absence of lowenergy threemagnon states.
Magnon decay channels for the extended model
In Model 2, the character of the quantum fluctuations away from zigzag order is notably different (Fig. 5). The finite Γ _{1} interaction reduces the local symmetry and leads to rotation of the ordered moments away from the cubic axes^{20, 37} (Fig. 5c). In the local picture, this allows additional singlespin fluctuations \(S_i^xS_j^z\left { \uparrow \uparrow } \right\rangle = \left { \downarrow \uparrow } \right\rangle\) (Fig. 5a), which correspond to oddorder anharmonic terms \({{\cal H}_3},{{\cal H}_5},...\) in the magnon Hamiltonian, where^{33, 34}:
At lowest order, such terms mix the singlemagnon states with the twomagnon continuum (ΔN _{tot} = ±1), via the scattering process depicted in Fig. 5b. The density of twomagnon states is shown in Fig. 5d, for the zigzag domain with Q = Y. In this case, at each kpoint the lowest energy twomagnon state \(a_{{{\bf{q}}_1}}^\dag a_{{{\bf{q}}_2}}^\dag \left 0 \right\rangle\) is obtained by placing one particle in the pseudoGoldstone mode at an Mpoint, and the second particle at q _{2} = k − M, with total energy \(E_2^{{\rm{min}}}({\bf{k}}) = {\epsilon _{{\bf{k}}  {\rm{M}}}} + {\epsilon _{\rm{M}}} \ne E_3^{{\rm{min}}}\). It should be emphasized that this condition differs from that of a conventional Heisenberg antiferromagnet (for which \(E_2^{{\rm{min}}} = E_3^{{\rm{min}}}\))^{34}. In the case of Model 2, the difference is directly related to the strong anisotropic K _{1} and Γ _{1} interactions, which shift the pseudoGoldstone modes to the Mpoints, such that only highenergy magnons remain at the Γpoint or ordering wavevector Q ^{38}. This shift therefore leads to an offset of the lowenergy even and odd magnon states in kspace such that \(E_2^{{\rm{min}}}({\bf{k}}) < {\epsilon _{{\bf{k}},1}}\) over a wide region of the Brillouin zone; there are many twomagnon states with equal or lower energy than the onemagnon states. Provided there is a finite Γ _{1}, the spontaneous decay of single magnons into the twoparticle continuum is therefore allowed even for the lowest magnon band. The decay rate is expected to be particularly large near the zone center, which represents a minimum in \(E_2^{{\rm{min}}}\). Similar kinematic conditions may also occur in other systems^{34, 39}. For Model 2, the pseudoGoldstone magnons near the Mpoints remain coherent due to the absence of lowenergy two particle states in which to decay (Fig. 5d). This explains the experimental observation of sharp magnonlike modes near the Mpoints^{24}. In contrast, the magnon bands in the remainder of the Brillouin zone directly overlap with the twoparticle continuum. It is therefore natural to anticipate a large decay rate even for the lowest magnon bands.
To confirm this idea, we have computed the threemagnon interactions and decay rates for all magnon bands for Model 2 using the selfconsistent imaginary Dyson equation (iDE) approach^{40}. Within this approach, it is assumed that the real part of the magnon selfenergy is already captured by the LSWT parameters, while the imaginary part is obtained selfconsistently (see “Methods” and Supplementary Note 3). The iDE approach therefore represents an extension of LSWT, in which the onemagnon excitations are broadened according to the momentum and banddependent decay rate γ _{ k,n }, while other contributions to the neutron intensity from multimagnon excitations are also absent^{41}. As a result, comparison of LSWT, ED, and iDE results (Fig. 6) allows for the identification of the origin of different contributions to the spectra.
The predicted neutron scattering intensity within the iDE approach (Fig. 6b) captures many of the most notable features that are observed in the ED and experimental data, showing a significant improvement over the LSWT results (Fig. 6a). First, there is an almost complete washout of the two highenergy onemagnon modes due to strong decays. This implies that the higherenergy features >4 meV appearing in ED are primarily multimagnon in character (including the 6 meV peak at the Γpoint). The appearance of these higherenergy features in the inelastic neutron response may arise partly from direct contributions from the broadened twomagnon continuum via the longitudinal component of the structure factor, which is not included in the iDE approach (see Supplementary Note 3). Second, the broadening of the two lower magnon bands in the iDE results and the resultant variation of their intensities are in a close agreement with the ED—particularly in a wide region near the Γpoint (see also Supplementary Fig. 5). These are precisely the features with which the LSWT results were most incompatible. Over much of the Brillouin zone—and especially for the higher magnon bands—the computed γ _{ k,n } is on the same scale as the onemagnon bandwidth, confirming the absence of coherent magnons.
Discussion
The general requirements for strong twomagnon decays are less restrictive than a proximity to a spin liquid state. Indeed, a large decay rate is ensured by the following three conditions: large anisotropic interactions, deviation of the ordered moments away from the highsymmetry axes, and strong overlap of the onemagnon states with the multimagnon continuum (see Supplementary Note 3). Of these, the first two conditions ensure that the scattering vertex \(\Lambda _{{\bf{12}}}^{\bf{3}}\) is large—of the order of the underlying interactions, i.e., \(\Lambda _{{\bf{12}}}^{\bf{3}}\sim {\cal O}\left( {{K_1},{\mathit \Gamma _1}} \right)\). For αRuCl_{3}, the strong overlap with the multimagnon continuum is ensured by shifting of the lowenergy magnons away from the Γpoint. Since the bottom of the twomagnon continuum must always have an energetic minimum at the Γpoint, the shifting of the pseudoGoldstone modes to a finite momentum ensures the remaining higherenergy magnons are degenerate with the continuum near the zone center. Experimentally, these conditions are also likely to be satisfied by the zigzag ordered Na_{2}IrO_{3} ^{9}, and spiral magnets αLi_{2}IrO_{3}, βLi_{2}IrO_{3}, and γLi_{2}IrO_{3} ^{42,43,44}. This picture is also consistent with recent indications that the magnetically disordered phase observed at high pressure in βLi_{2}IrO_{3} ^{45} is driven primarily by large Γ _{1} interactions^{46}.
With this in mind, there are two general scenarios that can explain the observed continuum excitations in αRuCl_{3} and the iridates A _{2}IrO_{3}. In the first scenario, which has been advanced by several studies, the excitations can be treated as free particles with a small number of flavors. Such excitations are weakly interacting and have welldefined dispersions, but possess quantum numbers (e.g., ΔS _{tot} = ±1/2) or topological properties inconsistent with the experimental neutron scattering selection rules (i.e., ΔS _{tot} = 0, ±1). The appearance of the broad continuum in energy therefore results only from the fact that these fractional excitations must be created in multiples. If they could have been created individually, they would have represented longlived and coherent quasiparticles with sharply peaked energies. This scenario indeed describes the Kitaev spin liquid, where the special symmetries of the Hamiltonian allow an exact description in terms of two flavors of particles: noninteracting Majorana spinons and localized fluxes^{5}. Such excitations are longlived, but belong to nontrivial topological sectors, and therefore cannot be created individually by any local operations. For the Kitaev spin liquid, the predicted continuum therefore represents coherent multiparticle excitations.
In contrast, upon moving away from the pure Kitaev point, the relevant symmetries that protect the spinons and fluxes are lifted both by additional magnetic interactions and by spontaneous symmetry breaking of the magnetic order. This tends to confine spinons into gauge neutral objects such as magnons^{47, 48}. Despite this latter tendency, we have argued that coherent magnons are unlikely to appear at large Γ _{1} due to the strong anharmonicity in the magnon Hamiltonian. While this leaves open the possibility that nearly free Majorana spinons persist into the zigzag ordered phase in some regions of the Brillouin zone, a more general scenario is that the observed continua represent fully incoherent excitations. In this second scenario, the excitations are not describable in terms of any type of free particles with small decay rates and welldefined dispersions. The broad continua therefore reflect the absence of coherent quasiparticles altogether, rather than particular experimental selection rules related to fractionalization. At present, it is not clear which of these scenarios applies to the iridates and αRuCl_{3}, although a key role must be played by both the Kitaev K _{1} and offdiagonal couplings such as Γ _{1}. In any case, the study of these materials calls into question the stability of magnetic quasiparticles in the presence of strongly anisotropic interactions.
In summary, we have shown that all main features of the magnetic excitations in αRuCl_{3} ^{12, 13, 24} are consistent with strongly anisotropic interactions having signs and relative magnitudes in agreement with ab initio predictions. The ferromagnetic Kitaev coupling (K _{1} < 0) is supplemented by a significant offdiagonal term (Γ _{1} > 0) that plays a crucial role in establishing both the zigzag order and the observed continua. In the presence of such interactions, the conventional magnon description breaks down even deep in the ordered phase, due to strong coupling of the onemagnon and twomagnon states. This effect is expected to persist over a large range of the phase diagram suggesting that the observed continua in αRuCl_{3} and the iridates A _{2}IrO_{3} represent a rich and general phenomenon extending beyond the Kitaev spin liquid. For this class of strongly spinorbitalcoupled magnets, the presence of complex and frustrated anisotropic interactions leads naturally to dominant anharmonic effects in the inelastic magnetic response. Fully describing the dynamics of these and similar materials therefore represents a formidable challenge that is likely to reveal aspects not found in conventional isotropic magnets.
Methods
Exact diagonalization
The neutron scattering intensity was computed via:
where f(k) is the atomic form factor of Ru^{3+} from ref. ^{49}. ED calculations were performed using the Lanczos algorithm^{50}, on several 20site and 24site clusters with periodic boundary conditions. Such periodic clusters are detailed in Supplementary Note 4. Excitations were computed using the continued fraction method^{51}. Further details and additional results are presented in the Supplementary Notes 4 and 5; these extensive calculations go beyond previous ED studies^{16, 17, 20, 26, 29}, which focused mainly on the static properties, or a limited portion of the phase diagram. ED results shown for the highsymmetry Γ, M, Y, X, and Γ′ points were averaged over all clusters. The ED kdependence of \({\cal I}({\bf{k}},\omega )\), integrated over the energy windows E = 1.3–2.3, 5.5–8.5, and 10.5+ meV (Figs. 2c, 3c), was obtained from a single 24site cluster respecting all symmetries of the model. The discrete ED spectra were Gaussian broadened by 0.5 meV, consistent with the width of experimental features^{12}. The intensities were also averaged over the same range of outofplane momentum as in the experiment^{12}.
Linear spinwave theory
LSWT results shown in Figs. 1 and 2 were obtained with the aid of SpinW^{52}. Following the approach with the ED data, the discrete LSWT spectra were as well Gaussian broadened by 0.5 meV and the intensities were also averaged over the same range of outofplane momentum as with ED and in the experiment^{12}.
Imaginary selfconsistent Dyson equation approach
In order to calculate magnon decay rates γ _{ k,n }, we have evaluated threemagnon interaction vertices by performing rotation to local reference frames of spins. The obtained value of the real–space interaction is quite large, about ~3 meV. Next, the Born approximation calculation of the decay rates results in unphysical divergencies^{34}, thus the selfenergy Σ_{ k,n } needs to be regularized. We have used the socalled iDE approach: a selfconsistent solution on the imaginary part of the Dyson’s equation, \({\Sigma _{{\bf{k}},n}}\left( {{\epsilon _{{\bf{k}},n}} + i{\gamma _{{\bf{k}},n}}} \right) =  i{\gamma _{{\bf{k}},n}}\), see ref. ^{40}. We have obtained the regularized broadening for the magnon spectrum and have calculated the transverse part of the dynamical structure factor, shown in Fig. 6, by adding the calculated decay rates to experimental resolution of 0.25 meV. The spectral function is approximated as a Lorentzian. More technical details can be found in the Supplementary Note 3.
Code availabilty
Custom computer codes used in this study are available from the corresponding author upon reasonable request. Documentation of the codes is not available.
Data availability
Data are available from the corresponding author upon reasonable request.
References
 1.
Lifshitz, E. M. & Pitaevskii, L. P. Statistical Physics Part 2 (Pergamon Press, Oxford, 1980).
 2.
Balents, L. Spin liquids in frustrated magnets. Nature 464, 199–208 (2010).
 3.
Coldea, R., Tennant, D. A., Tsvelik, A. M. & Tylczynski, Z. Experimental realization of a 2D fractional quantum spin liquid. Phys. Rev. Lett. 86, 1335–1338 (2001).
 4.
Han, T.H. et al. Fractionalized excitations in the spinliquid state of a kagomelattice antiferromagnet. Nature 492, 406–410 (2012).
 5.
Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006).
 6.
Jackeli, G. & Khaliullin, G. Mott insulators in the strong spinorbit coupling limit: from Heisenberg to a quantum compass and Kitaev models. Phys. Rev. Lett. 102, 017205 (2009).
 7.
Rau, J. G., Lee, E. K.H. & Kee, H.Y. Spinorbit physics giving rise to novel phases in correlated systems: iridates and related materials. Annu. Rev. Condens. Matter Phys. 7, 195–221 (2016).
 8.
Gretarsson, H. et al. Magnetic excitation spectrum of Na_{2}IrO_{3} probed with resonant inelastic Xray scattering. Phys. Rev. B 87, 220407 (2013).
 9.
Hwan Chun, S. et al. Direct evidence for dominant bonddirectional interactions in a honeycomb lattice iridate Na_{2}IrO_{3}. Nat. Phys. 11, 462–466 (2015).
 10.
Sandilands, L. J., Tian, Y., Plumb, K. W., Kim, Y.J. & Burch, K. S. Scattering continuum and possible fractionalized excitations in αRuCl_{3}. Phys. Rev. Lett. 114, 147201 (2015).
 11.
Nasu, J., Knolle, J., Kovrizhin, D. L., Motome, Y. & Moessner, R. Fermionic response from fractionalization in an insulating twodimensional magnet. Nat. Phys. 12, 912–915 (2016).
 12.
Banerjee, A. et al. Neutron scattering in the proximate quantum spin liquid αRuCl_{3}. Science 356, 1055–1059 (2017).
 13.
Banerjee, A. et al. Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet. Nat. Mater. 15, 733–740 (2016).
 14.
Do, S.H. et al. Incarnation of Majorana fermions in Kitaev quantum spin lattice. Preprint at https://arxiv.org/abs/1703.01081 (2017).
 15.
Glamazda, A., Lemmens, P., Do, S.H., Choi, Y. S. & Choi, K.Y. Raman spectroscopic signature of fractionalized excitations in the harmonichoneycomb iridates β and γLi_{2}IrO_{3}. Nat. Commun. 7, 12286 (2016).
 16.
Chaloupka, J., Jackeli, G. & Khaliullin, G. KitaevHeisenberg model on a honeycomb lattice: possible exotic phases in iridium oxides A _{2}IrO_{3}. Phys. Rev. Lett. 105, 027204 (2010).
 17.
Rau, J. G., Lee, E. K.H. & Kee, H.Y. Generic spin model for the honeycomb iridates beyond the Kitaev limit. Phys. Rev. Lett. 112, 077204 (2014).
 18.
Winter, S. M., Li, Y., Jeschke, H. O. & Valent, R. Challenges in design of Kitaev materials: magnetic interactions from competing energy scales. Phys. Rev. B 93, 214431 (2016).
 19.
Sears, J. A. et al. Magnetic order in αRuCl_{3}: a honeycomblattice quantum magnet with strong spinorbit coupling. Phys. Rev. B 91, 144420 (2015).
 20.
Chaloupka, J. & Khaliullin, G. Magnetic anisotropy in the Kitaev model systems Na_{2}IrO_{3} and RuCl_{3}. Phys. Rev. B 94, 064435 (2016).
 21.
Kimchi, I., Coldea, R. & Vishwanath, A. Unified theory of spiral magnetism in the harmonichoneycomb iridates α, β, and γLi_{2}IrO_{3}. Phys. Rev. B 91, 245134 (2015).
 22.
Kim, H.S., Lee, E. K.H. & Kim, Y.B. Predominance of the Kitaev interaction in a threedimensional honeycomb iridate: from ab initio to spin model. EPL 112, 67004 (2015).
 23.
Cao, H. B. et al. Lowtemperature crystal and magnetic structure of αRuCl_{3}. Phys. Rev. B 93, 134423 (2016).
 24.
Ran, K. et al. Spinwave excitations evidencing the Kitaev interaction in single crystalline αRuCl_{3}. Phys. Rev. Lett. 118, 107203 (2017).
 25.
Kim, H.S. & Kee, H.Y. Crystal structure and magnetism in αRuCl_{3}: an ab initio study. Phys. Rev. B 93, 155143 (2016).
 26.
Yadav, R. et al. Kitaev exchange and fieldinduced quantum spinliquid states in honeycomb αRuCl_{3}. Sci. Rep. 6, 37925 (2016).
 27.
Wang, W., Dong, Z.Y., Yu, S.L. & Li, J.X. Theoretical investigation of the magnetic dynamics and superconducting pairing symmetry in αRuCl_{3}. Phys. Rev. B 96, 115103 (2017).
 28.
Hou, Y. S., Xiang, H. J. & Gong, X. G. Unveiling magnetic interactions of ruthenium trichloride via constraining direction of orbital moments: potential routes to realize a quantum spin liquid. Phys. Rev. B 96, 054410 (2017).
 29.
Katukuri, V. M. et al. Kitaev interactions between j = 1/2 moments in honeycomb Na_{2}IrO_{3} are large and ferromagnetic: insights from ab initio quantum chemistry calculations. New J. Phys. 16, 013056 (2014).
 30.
Gohlke, M., Verresen, R., Moessner, R. & Pollmann, F. Dynamics of the KitaevHeisenberg model. Preprint at https://arxiv.org/abs/1701.04678 (2017).
 31.
Johnson, R. D. et al. Monoclinic crystal structure of αRuCl_{3} and the zigzag antiferromagnetic ground state. Phys. Rev. B 92, 235119 (2015).
 32.
Harris, A. B., Kumar, D., Halperin, B. I. & Hohenberg, P. C. Dynamics of an antiferromagnet at low temperatures: spinwave damping and hydrodynamics. Phys. Rev. B 3, 961–1024 (1971).
 33.
Chernyshev, A. L. & Zhitomirsky, M. E. Magnon decay in noncollinear quantum antiferromagnets. Phys. Rev. Lett. 97, 207202 (2006).
 34.
Zhitomirsky, M. E. & Chernyshev, A. L. Colloquium: spontaneous magnon decays. Rev. Mod. Phys. 85, 219–242 (2013).
 35.
Ruvalds, J. & Zawadowski, A. Twophonon resonances and hybridization of the resonance with singlephonon states. Phys. Rev. B 2, 1172–1175 (1970).
 36.
Kosevich, A. M. The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices (WileyVCH Verlag GmbH & Co. KGaA, Weinheim, 2005).
 37.
Sizyuk, Y., Wölfle, P. & Perkins, N. B. Selection of direction of the ordered moments in Na_{2}IrO_{3} and αRuCl_{3}. Phys. Rev. B 94, 085109 (2016).
 38.
Chaloupka, J. & Khaliullin, G. Hidden symmetries of the extended KitaevHeisenberg model: implications for the honeycomblattice iridates A _{2}IrO_{3}. Phys. Rev. B 92, 024413 (2015).
 39.
Hwang, K. & Kim, Y. B. Theory of triplon dynamics in the quantum magnet BiCu_{2}PO_{6}. Phys. Rev. B 93, 235130 (2016).
 40.
Maksimov, P. A., Zhitomirsky, M. E. & Chernyshev, A. L. Fieldinduced decays in XXZ triangularlattice antiferromagnets. Phys. Rev. B 94, 140407 (2016).
 41.
Mourigal, M., Fuhrman, W. T., Chernyshev, A. L. & Zhitomirsky, M. E. Dynamical structure factor of the triangularlattice antiferromagnet. Phys. Rev. B 88, 094407 (2013).
 42.
Biffin, A. et al. Noncoplanar and counterrotating incommensurate magnetic order stabilized by Kitaev interactions in γLi_{2}IrO_{3}. Phys. Rev. Lett. 113, 197201 (2014).
 43.
Biffin, A. et al. Unconventional magnetic order on the hyperhoneycomb Kitaev lattice in βLi_{2}IrO_{3}: full solution via magnetic resonant Xray diffraction. Phys. Rev. B 90, 205116 (2014).
 44.
Williams, S. C. et al. Incommensurate counterrotating magnetic order stabilized by Kitaev interactions in the layered honeycomb αLi_{2}IrO_{3}. Phys. Rev. B 93, 195158 (2016).
 45.
Takayama, T. et al. Hyperhoneycomb iridate βLi_{2}IrO_{3} as a platform for Kitaev magnetism. Phys. Rev. Lett. 114, 077202 (2015).
 46.
Kim, H.S., Kim, Y. B. & Kee, H.Y. Revealing frustrated local moment model for pressurized hyperhoneycomb iridate: paving the way toward a quantum spin liquid. Phys. Rev. B 94, 245127 (2016).
 47.
Mandal, S., Bhattacharjee, S., Sengupta, K., Shankar, R. & Baskaran, G. Confinementdeconfinement transition and spin correlations in a generalized Kitaev model. Phys. Rev. B 84, 155121 (2011).
 48.
Schaffer, R., Bhattacharjee, S. & Kim, Y.B. Quantum phase transition in HeisenbergKitaev model. Phys. Rev. B 86, 224417 (2012).
 49.
Cromer, D. T. & Waber, J. T. Scattering factors computed from relativistic DiracSlater wave functions. Acta Cryst. 18, 104–109 (1965).
 50.
Lanczos, C. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Nat. Bur. Stand. 45, 255–282 (1950).
 51.
Dagotto, E. Correlated electrons in hightemperature superconductors. Rev. Mod. Phys. 66, 763–840 (1994).
 52.
Toth, S. & Lake, B. Linear spinwave theory for singleQ incommensurate magnetic structures. J. Phys. Condens. Matter 27, 166002 (2015).
 53.
Knolle, J., Kovrizhin, D. L., Chalker, J. T. & Moessner, R. Dynamics of fractionalization in quantum spin liquids. Phys. Rev. B 92, 115127 (2015).
 54.
Knolle, J., Kovrizhin, D. L., Chalker, J. T. & Moessner, R. Dynamics of a twodimensional quantum spin liquid: signatures of emergent Majorana fermions and fluxes. Phys. Rev. Lett. 112, 207203 (2014).
Acknowledgements
The authors acknowledge useful discussions with J. Chaloupka, A. Banerjee, S.E. Nagler, A.A. Tsirlin, R. Moessner, F. Pollmann, and M. Zhitomirsky. S.M.W. acknowledges support through an NSERC Canada Postdoctoral Fellowship. R.V. and K.R. acknowledge support by the Deutsche Forschungsgemeinschaft through grant SFB/TR 49. The work of P.A.M. and A.L.C. was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under award no. DEFG0204ER46174.
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R.V. and S.M.W. conceived the project. K.R., S.M.W. and A.H. performed and analyzed the ED calculations. P.A.M. and A.L.C. performed and analyzed the iDE results. All authors contributed equally to the manuscript.
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Correspondence to Stephen M. Winter.
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Winter, S.M., Riedl, K., Maksimov, P.A. et al. Breakdown of magnons in a strongly spinorbital coupled magnet. Nat Commun 8, 1152 (2017). https://doi.org/10.1038/s41467017011770
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