Abstract
The microscopic origins of emergent behaviours in condensed matter systems are encoded in their excitations. In ordinary magnetic materials, single spinflips give rise to collective dipolar magnetic excitations called magnons. Likewise, multiple spinflips can give rise to multipolar magnetic excitations in magnetic materials with spin S ≥ 1. Unfortunately, since most experimental probes are governed by dipolar selection rules, collective multipolar excitations have generally remained elusive. For instance, only dipolar magnetic excitations have been observed in isotropic S = 1 Haldane spin systems. Here, we unveil a hidden quadrupolar constituent of the spin dynamics in antiferromagnetic S = 1 Haldane chain material Y_{2}BaNiO_{5} using Ni L_{3}edge resonant inelastic xray scattering. Our results demonstrate that pure quadrupolar magnetic excitations can be probed without direct interactions with dipolar excitations or anisotropic perturbations. Originating from onsite double spinflip processes, the quadrupolar magnetic excitations in Y_{2}BaNiO_{5} show a remarkable dual nature of collective dispersion. While one component propagates as noninteracting entities, the other behaves as a bound quadrupolar magnetic wave. This result highlights the rich and largely unexplored physics of higherorder magnetic excitations.
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Introduction
The elementary excitation of a magnetically ordered material is a single dipolar spinflip of an electron, delocalised coherently across the system in the form of a spin wave. The spinwave quasiparticle, known as a magnon, carries a spin angular momentum of one unit and has welldefined experimental signatures. Collective dipolar magnetic excitations also appear in lowdimensional magnets that remain disordered to the lowest achievable temperatures because of quantum fluctuations. A paradigmatic example is the S = 1 antiferromagnetic Haldane spin chain, where magnetic order is suppressed in favour of a singlet ground state with nonlocal topological order^{1,2}. Several theoretical and experimental works have established that a single spinflip from this exotic ground state creates dipolar magnetic excitations that propagate along the chain above an energy gap of Δ_{H} ~ 0.41J, the Haldane gap^{3,4,5,6,7,8}.
Materials hosting 3d transition metal ions with a d^{2} or d^{8} configuration (such as Ni^{2+} for the latter) often possess strongly interacting spin1 local magnetic moments with quantised spin projections S_{z} = −1, 0, 1. In addition to the usual single spinflip excitations, it is possible to create quadrupolar magnetic excitations by changing the composite spin by two units. Such excitations can be conceived as flipping two of the constituent spin1/2’s, as shown in Fig. 1a. Incidentally, quadrupolar magnetic waves arising from such transitions and carrying two units of angular momentum were predicted for S = 1 ferromagnetic chains as early as the 1970s^{9,10} and may play a role in the iron pnictide superconductors^{11}. Since most probes are restricted by dipolar selection rules, however, such quadrupolar excitations have largely evaded detection except in rare situations where they are perturbed by anisotropic interactions, spinorbit coupling, lattice vibrations, or large magnetic fields^{12,13,14,15,16,17}. Quadrupolar magnetic excitations have never been observed in isotropic Haldane spin chains, even though dipolar magnetic excitations have been extensively studied^{3,4,5,6,7,8,18}. It is then natural to wonder whether purely quadrupolar collective magnetic excitations exist in isotropic spin1 systems.
Here, we uncover the presence of collective quadrupolar magnetic excitations in the isotropic S = 1 Haldane chain system Y_{2}BaNiO_{5} using high energyresolution Ni L_{3}edge resonant inelastic xray scattering (RIXS). Previous studies on nickelates have already shown that Ni L_{3}edge RIXS can probe dipolar magnons^{19,20}. In addition, ref. ^{21} recently showed that double spinflips are allowed at this edge through the combined manybody effect of corevalence exchange and corehole spinorbit interactions^{22,23} (Fig. 1b), making it the optimal tool for this study. Y_{2}BaNiO_{5} is one of the best realisations of the isotropic Haldane spin chain material with intrachain exchange J ~ 24 meV and negligibly small singleion anisotropy ~0.035J, exchangeanisotropy ~0.011J, and interchain exchange ~0.0005J^{6,24}. This aspect allows us to describe the system completely using a simple Heisenberg model, and emphasises the relevance of the pure quadrupolar magnetic excitations in the spin dynamics of S = 1 systems.
Results
Excitations in Haldane spin chains
In a valence bond singlet scheme, the Heisenberg model’s ground state in the Haldane phase can be represented as a macroscopic S_{tot} = 0 state comprised of pairs of fictitious spin\(\frac{1}{2}\) particles on neighbouring sites that form antisymmetric singlets on each bond^{8} (Fig. 1c). A single local spinflip breaks a bond singlet to form a triplet excitation, raising the chain’s total spin quantum number to S_{tot} = 1. In contrast, a local double spinflip would disrupt singlets on either side of the excited site, creating a pair of triplet excitations and raising the spin quantum number to S_{tot} = 2. In the RIXS process, it is also possible to have a total spinconserved twosite excitation with ΔS_{tot} = 0 (see Fig. 1c), which appears as a twomagnon continuum. To simplify our notation, we will refer to the singlesite singlespinflip induced dipolar ΔS_{tot} = 1 excitation as ΔS_{1}, the singlesite doublespinflip induced quadrupolar ΔS_{tot} = 2 excitation as ΔS_{2}, and the twosite ΔS_{tot} = 0 twomagnon excitation as ΔS_{0}.
Figure 2a shows a RIXS intensity map collected on Y_{2}BaNiO_{5} (see Methods). The feature with the largest spectral weight follows a dispersion relation consistent with inelastic neutron scattering (INS) results for the dipolar ΔS_{1} excitation^{4,5,24}. This feature is also reproduced in our density matrix renormalisation group (DMRG) calculations of the dynamical structure factor S_{1}(q_{∥}, ω) for an isotropic Heisenberg model (see Fig. 2b and Methods). Ni L_{3}edge RIXS is unable to reach the exact antiferromagnetic zone center (q_{∥} = 0.5, in units of 2π/c throughout), where the Haldane gap of ~ 8.5 meV exists. But it does probe an equally interesting region close to q_{∥} = 0. Prior work^{4,7,25} has focused on observing the breakdown of the welldefined ΔS_{1} quasiparticle into a twomagnon continuum for q_{∥} ≾ 0.12 and the spectral weight vanishing as \({q}_{\parallel }^{2}\). Although we notice a reduction in the intensity of the ΔS_{1} excitation in our experiment, we also observe significant inelastic spectral weight near zero energy close to q_{∥} = 0 (also see Fig. 2e for the line spectrum at q_{∥} = 0.01). Interestingly, a new dispersing excitation is clearly visible with an energy maximum of ~ 136 meV at q_{∥} = 0. Of the calculated S_{0}(q_{∥}, ω), S_{1}(q_{∥}, ω), S_{2}(q_{∥}, ω) dynamical structure factors, shown in Fig. 2d, b, c, respectively, only S_{2}(q_{∥}, ω) has spectral weight in this region. Moreover, the S_{2}(q_{∥} ~ 0, ω) line profile has two components: a broad lowenergy continuum and a sharp highenergy peak, as shown in Fig. 2f. It therefore appears that both the low and highenergy spectral components seen in the experiment can be described by the quadrupolar ΔS_{2} excitation in this region of momentum space.
To evaluate the relative contributions of each excitation, we decomposed the RIXS spectra across the momentum space into the dynamical structure factors calculated by DMRG. Figure 3a–c show representative RIXS spectra. The data are fitted with an elastic peak and experimental energyresolution convoluted line profiles of S_{α}(q_{∥}, ω) (α = 0, 1, 2, see Methods). While the spectra are dominated by the dipolar ΔS_{1} excitations at high q_{∥} values, the spectral weight close to q_{∥} = 0 can be fitted with only the quadrupolar ΔS_{2} excitation. Figure 3d and e show the RIXS spectra after subtracting other fitted components to keep only the ΔS_{1} and ΔS_{2} contributions, respectively, along with their fitted profiles from DMRG. This analysis shows that the lowenergy RIXS spectral weight is carried by the lowenergy continuum component of the ΔS_{2} excitations below the quasiparticle decay threshold momentum. This component is gapped and has a peak energy of ~19 meV at q_{∥} ~ 0, reminiscent of the lower boundary of the twomagnon continuum at 2Δ_{H} predicted for Haldane spin chains^{7}. We note that the energy gap at this momentum has not yet been confirmed for any Haldane spin chain by INS due to the small scattering crosssections. Figure 3f shows the RIXS intensity map after subtracting only the elastic peaks and Fig. 3g shows the combined DMRG dynamical structure factors S_{0}(q_{∥}, ω), S_{1}(q_{∥}, ω), and S_{2}(q_{∥}, ω) obtained by fitting the RIXS spectra. The ΔS_{0} type of twomagnon continuum excitations provide a negligible contribution to the RIXS spectra (see Methods for the contributions from each type of excitation). Undoubtedly, the twocomponent excitation (the broad lowenergy continuum and the sharp highenergy component), and its dispersion in the momentum space (Fig. 3e) is well described only by the quadrupolar ΔS_{2} excitation.
Quadrupolar excitations at finite T
To further understand the character of the twocomponent ΔS_{2} quadrupolar excitations, we also studied their thermal evolution. Figure 4a shows the RIXS spectra at q_{∥} = 0.01, with contributions from the ΔS_{2} excitations, for increasing temperatures. For comparison, Fig. 4b shows finite temperature DMRG simulations for the S_{2}(q_{∥} = 0.01, ω) excitations, convoluted with the experimental energy resolution. Since the raw DMRG data for the ΔS_{2} channel (Fig. 2f) contains a twopeak structure with an asymmetric peak appearing at low energy and a symmetric peak at high energy, we fit the experimental S_{2}(q_{∥} = 0.01, ω) finite temperature data using two components (see methods). Our results show that only the ΔS_{2} excitations obtained from DMRG are needed to reproduce the experimental RIXS spectra, even at finite temperatures. As shown in Fig. 4c, the peak energy of the lower component increases with temperatures following twice of the system’s Haldane gap from ref. ^{5}. Conversely, the highenergy peak begins to soften above the Haldane gap temperature of ~ 100 K. The highest energy value of the ΔS_{1} peak at q_{∥} = 0.25 also follows this trend. The bandwidth reduction of the ΔS_{1} triplet dispersion with temperature occurs due to the thermal blocking of propagation lengths and decoherence^{26}. In a simple picture, if one considers a continuum from pairs of noninteracting triplets due to single spinflips at multiple sites, then bandwidth reduction of each would manifest as the overall raising and lowering of the lower and upper boundaries of the continuum, respectively (see Supplementary Note 3). A similar thermal effect on the propagation of the quadrupolar ΔS_{2} excitation should occur. The spectral weight of the two components in the ΔS_{2} excitation, however, behave differently with temperature. The lowenergy continuum intensity varies little, while the highenergy peak diminishes rapidly with increasing temperature and the rate of decay is comparable to the ΔS_{1} peak at q_{∥} = 0.47 (see Fig. 4d). The DMRG calculated correlation lengths (in lattice units) of the ΔS_{1} and ΔS_{2} excitations, as shown in Fig. 4e, also decrease in a similar way with temperature.
Dual nature of the quadrupolar excitations
The energymomentum and temperature dependence of the quadrupolar ΔS_{2} excitations implies that their low and highenergy components are different in nature. At lowenergy, a conceivable picture is that immediately after a pair of triplets are created by a singlesite excitation, they decay into two noninteracting triplets (see Fig. 1d) propagating incoherently along the chain and giving rise to the broad lowenergy continuum (see Fig. 2c for the expected continuum boundaries). This behaviour is remarkably similar to the fractionalisation of a ΔS_{tot} = 1 excitation into a twospinon continuum in isotropic spin1/2 chains^{27}, but having a distinct origin.
The sharpness of the highenergy component and its rapid decay with temperature, on the other hand, hints that it behaves as a distinct quasiparticle formed from pairs of triplets propagating coherently (see Fig. 1d)^{28,29}. In lowdimensional systems, sharp peaks in the magnetic spectrum may either originate from a van Hove singularity in the density of states of quasiparticles (in our case, noninteracting triplet pairs) or from the formation of a bound state (in our case, bound triplet pairs). In Fig. 2c, the lower and upper boundaries of the continuum from pairs of noninteracting triplets (equivalent to the twomagnon continuum) are shown. The highenergy component of ΔS_{2} appears above the upper boundary of the continuum, ruling out the van Hove singularity scenario and suggesting the formation of a bound state. The peak energy of the highenergy component is slightly larger than twice the highest energy value of the ΔS_{1} peak (by ~ 7 meV at T = 11 K) and, surprisingly, remains so up to the highest measured temperature. The small positive energy difference suggests a weak repulsive interaction between the bound triplets formed after a quadrupolar ΔS_{2} excitation^{9,30}. Supplementary Note 6 provides a semiquantitative energy scale based argument to support the notion of the bound state of weakly repulsing triplets excitations.
The correlation length of the ΔS_{1} excitations in Haldane spin chains decay exponentially in the presence of a nonlocal order, which can be viewed to originate from alternating S_{z} = ±1 sites intervened by S_{z} = 0 sites. The ΔS_{2} excitations also have an exponentially decaying correlation length, albeit smaller than ΔS_{1} excitations. The difference in correlation length is likely due to the fact that the creation of quadrupolar wave excitations costs energetically at least twice as much the single spinflip magnonlike excitations at any wavevector. As such, they may provide a means to detect the hidden nonlocal order that, at present, is only estimated theoretically by considering the S_{z} = ±1 states in Haldane spin chains^{2,31,32}. Overall, we show that the ΔS_{2} excitations sustain the isotropic properties, the exchange interactions, and the coherence inherent to the system.
Discussion and outlook
Magnetic excitations provide vital information about a system’s thermodynamic, magnetotransport, ultrafast magnetic, spintronic, or superconducting properties. Moreover, higherorder multipolar spin and orbital degrees of freedom give rise to exotic nonclassical phenomena like the Kitaev spin liquid^{33}, multispinons^{34}, spinnematicity^{15}, onsite multiferroicity^{12}, and boundmagnon states ^{14,16}, in a wide variety of magnetic systems. However, as noted earlier, multipolar excitations are challenging to detect using conventional probes^{35}. It was shown recently that quadrupolar excitations in S = 1 FeI_{2} appear in INS only due to their hybridisation with dipolar excitations through anisotropic spinexchanges^{14}. In the presence of a strong anisotropy, bound magnetic excitations from ΔS_{tot} = 2 spinflips at energies higher than the twomagnon continuum have also been observed in S = 1 spin chains using highfield electron spin resonance^{36,37}. However, our demonstration of pure quadrupolar spin dynamics in an isotropic Haldane system, without invoking anistropic interactions, suggests that simultaneous confinement and propagation of excitations can occur entirely via higherorder quantum correlations^{38}. This work thus illustrates RIXS’s capability of detecting higherorder dispersing excitations, irrespective of the presence of dipolar excitations^{14}, and thus may be the preferable way to study quadrupolar excitations in, for instance, spinnematic systems, where the dipolar excitations are suppressed^{39,40}. Also recently, it has been seen that Cu L_{3}edge RIXS can probe spinconserved and nonspinconserved higherorder fourspinons in spin1/2 Heisenberg antiferromagnetic spin chains^{41}. Our present work further consolidates Ledge RIXS (in comparison to O Kedge)^{34}, as a powerful probe for characterising nonlocal longrange magnetic correlations via the study of higherorder spinflip excitations, thereby, extending both the energymomentum phase space and the diversity of the magnetic quantum systems that can be explored. On the other hand, exploring the physics of highenergy excitations and/or eigenstates of a simple nonintegrable spin chain model is in itself of great theoretical interest. We provide a simple physical intuition about the nature of one of the highenergy eigenstates of the onedimensional Heisenberg model and our findings may have important consequences for manybody correlated states of matter and thermalisation in quantum systems^{42}. Looking forward, it would be interesting to learn how the quadrupolar excitations can be manipulated with intrinsic perturbations like anisotropy or extrinsic ones like a magnetic field. The indications of a propagating quadrupolar bound excitation in a real material can also have important ramifications for realising quantum information transfer in form of qubit pairs^{43,44}.
Methods
Experiments
A single crystal of Y_{2}BaNiO_{5} grown by the floatingzone method was used for the RIXS measurements. The momentum transfer along the chain direction q_{∥} was varied by changing the xray incident angle θ while keeping the scattering angle fixed at 154^{∘}. The lattice constant along the chain or the caxis used for the calculation of momemtum transfer is 3.77 Å. The crystal was cleaved in vacuum and the pressure in the experimental chamber was maintained below ~5 × 10^{−10} mbar throughout the experiment. High energyresolution RIXS data (ΔE ≃ 37 meV) at the Ni L_{3}edge were collected at I21 RIXS beamline, Diamond Light Source, United Kingdom^{45}. The zeroenergy position and resolution of the RIXS spectra were determined from subsequent measurements of elastic peaks from an adjacent carbon tape. The polarization vector of the incident xray was parallel to the scattering plane (i.e. π polarization). See Supplementary Note 1 for more details of the experimental configuration.
Theory
In the main text, we pointed out that the S = 1 Haldane chain system Y_{2}BaNiO_{5} might present negligibly small singleion anisotropy ~0.035J, exchangeanisotropy ~0.011J terms in a Heisenberg model description at low energies. We have verified numerically that these small corrections do not change our magnetic spectra qualitatively, and therefore a pure isotropic Heisenberg model has been adopted throughout our study.
Zero and finite temperature DMRG Calculations
T = 0 DMRG calculations on 100 site chains with open boundary conditions (OBC) were carried out with the correctionvector method^{46} using the Krylov decomposition^{47}, as implemented in the DMRG++ code^{48}. This approach requires realspace representation for the dynamical structure factors in the frequency domain, which can be found in the Supplementary Note 4. For T > 0 calculations we used the ancilla (or purification) method with a system of 32 physical and 32 ancilla sites, also with OBC. For more details see Supplementary Note 7.
For both the zero temperature and finite temperature calculations we kept up to m = 2000 DMRG states to maintain a truncation error below 10^{−7} and 10^{−6}, respectively and introduced a spectral broadening in the correctionvector approach fixed at η = 0.25J = 6 meV.
Dynamical spin correlation functions
We consider three correlation functions S_{0}(q_{∥}, ω), S_{1}(q_{∥}, ω), and S_{2}(q_{∥}, ω) giving information about ΔS_{tot} = 0, ΔS_{tot} = 1, and ΔS_{tot} = 2 excitations, respectively. To make the expressions more transparent, we use the Lehmann representation and construct the corresponding excitation operators in momentum space. The relevant correlation functions are
where \(\leftf\right\rangle\) are the final states of the RIXS process and
The three dynamical correlation functions given by Eq. (1) appear at the lowest order of a ultrashort corehole lifetime expansion of the full RIXS crosssection. As the singlesite ΔS_{tot} = 0 RIXS scattering operator is trivial (identity operator) in a lowenergy description in terms of spin S = 1 sites, the lowest order operator would involve twosites and, by rotational symmetry, involves a scalar product of neighbouring spin operators [see Eq. (2)]. Single and double spinflip RIXS scattering operators, on the other hand, lead to ΔS_{tot} = 1 and ΔS_{tot} = 2 excitations and can be naturally described in terms of onsite \({S}_{j}^{+}\) and \({({S}_{j}^{+})}^{2}\) operators, respectively [Eqs. (3) & (4)]. In the Supplementary Note 5 we provide analysis of the three dynamical correlation functions in terms of single triplet excitations or magnon states in the Haldane chain.
Correlation lengths
Figure 4e of the main text shows dipolar and quadrupolar correlation lengths as a function of temperature. These have been obtained by computing \(\left\langle \psi (\beta )\right{S}_{j}^{+}{S}_{j+r}^{}\left\psi (\beta )\right\rangle\) and \(\left\langle \psi (\beta )\right{({S}_{j}^{+})}^{2}{({S}_{j+r}^{})}^{2}\left\psi (\beta )\right\rangle\) correlation functions from the center of the chain j = c, respectively, and fitting with a exponential decay relationship f(r) = Ae^{−r/ξ}.
RIXS data fitting
RIXS data were normalised to the incident photon flux and corrected for xray selfabsorption effects prior to fitting. The elastic peak was fit with a Gaussian function with a width set by the energy resolution. The RIXS spin excitations in Fig. 3 were modeled with the Bose factor weighted dynamical spin susceptibilities obtained from our T = 0 DMRG calculations for ΔS_{tot} = 0, 1, and 2 (S_{0}, S_{1} and S_{2}, respectively) after they were convoluted with a Gaussian function capturing the experimental energy resolution. The total model intensity is given by
where the coefficients C_{0}, C_{1} and C_{2} account for the varying RIXS scattering cross section for each spin excitation with varying θ (q_{∥}). The reader is referred to the Supplementary Note 2 for the extracted values of coefficients and the fit profiles. As seen in Fig. 3c, S_{2} from DMRG for ΔS_{tot} = 2 transitions do not capture the additional spectral weight on the high energy side in the RIXS signal. A ‘half’Lorentzian truncated damped harmonic oscillator (HLDHO) function centered at 0.175 eV was therefore included in the fits to account for this tail. In Fig. 4, the ΔS_{tot} = 2 excitations at q_{∥} = 0.01(2π/c) are decomposed using a DHO function for the sharp high energy peak and a skewed Gaussian function for the broad lower continuum. Both functions are energyresolution convoluted and weighted by a Bose factor. DHO functions were used similarly for fitting ΔS_{tot} = 1 triplet excitations at q_{∥} = 0.47 (see Supplementary Note 2).
Data availability
The instruction to build the input scripts for the DMRG++ package to reproduce our results can be found in the Supplementary Material. The data to reproduce our figures is available as a public data set at https://doi.org/10.5281/zenodo.6394852. Raw data files will be made available upon request.
Code availability
The numerical results reported in this work were obtained with DMRG++ versions 6.01 and PsimagLite versions 3.01. The DMRG++ computer program^{48} is available at https://github.com/g1257/dmrgpp.git (see Supplementary Note 7 for more details.)
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Acknowledgements
We thank I. Affleck, J.G. Park, S. Hayden, A. Aligia, and K. Wohlfeld for insightful discussions. We are grateful to C. Bastista for suggesting the possibility of a van Hove singularity in our results. All data were taken at the I21 RIXS beam line of Diamond Light Source (United Kingdom) using the RIXS spectrometer designed, built, and owned by Diamond Light Source. We acknowledge Diamond Light Source for providing the beam time on beam line I21 under Proposal MM24593. S. J. acknowledges support from the National Science Foundation under Grant No. DMR1842056. A. Nocera acknowledges support from the Max PlanckUBCUTokyo Center for Quantum Materials and Canada First Research Excellence Fund (CFREF) Quantum Materials and Future Technologies Program of the Stewart Blusson Quantum Matter Institute (SBQMI), and the Natural Sciences and Engineering Research Council of Canada (NSERC). S.W.C. was supported by the DOE under Grant No. DOE: DEFG0207ER46382. This work used computational resources and services provided by Compute Canada and Advanced Research Computing at the University of British Columbia. We acknowledge T. Rice for the technical support throughout the beam times. We also thank G. B. G. Stenning and D. W. Nye for help on the Laue instrument in the Materials Characterisation Laboratory at the ISIS Neutron and Muon Source.
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K.J.Z. conceived the project; K.J.Z., A. Nag, A. Nocera, and S.J. supervised the project. A. Nag, K.J.Z., S.A., M.G.F., and A.C.W. performed RIXS measurements. A. Nag, S.A., and K.J.Z. analysed RIXS data. S.W.C. synthesized and characterised the sample. A. Nocera and S.J. performed DMRG calculations. A. Nag, K.J.Z., A. Nocera, and S.J. wrote the manuscript with comments from all the authors.
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Nag, A., Nocera, A., Agrestini, S. et al. Quadrupolar magnetic excitations in an isotropic spin1 antiferromagnet. Nat Commun 13, 2327 (2022). https://doi.org/10.1038/s41467022300655
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DOI: https://doi.org/10.1038/s41467022300655
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