Abstract
Onedimensional (1D) magnetic insulators have attracted significant interest as a platform for studying quasiparticle fractionalization, quantum criticality, and emergent phenomena. The spin1/2 Heisenberg chain with antiferromagnetic nearest neighbour interactions is an important reference system; its elementary magnetic excitations are spin1/2 quasiparticles called spinons that are created in even numbers. However, while the excitation continuum associated with twospinon states is routinely observed, the study of fourspinon and higher multispinon states is an open area of research. Here we show that fourspinon excitations can be accessed directly in Sr_{2}CuO_{3} using resonant inelastic xray scattering (RIXS) in a region of phase space clearly separated from the twospinon continuum. Our finding is made possible by the fundamental differences in the correlation function probed by RIXS in comparison to other probes. This advance holds promise as a tool in the search for novel quantum states and quantum spin liquids.
Introduction
When confined to one spatial dimension (1D), systems of interacting electrons host an assortment of macroscopic manybody phenomena, including quantum critical magnetic states with collective excitations carrying fractional quantum numbers. For this reason, quasi1D magnetic insulators have attracted wide experimental and theoretical interest as an ideal playground for studying quantum manybody phenomena. Owing to numerous experimental realizations of such models in real materials, some of the most stringent tests of quantum manybody theory have been conducted in 1D^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19}.
The 1D Heisenberg antiferromagnet (HAFM), where localized spins S interact with their nearest neighbours via an exchange interaction J, is perhaps the simplest and best understood of these systems; the spin1/2 case is an important reference system that can be solved exactly using the Bethe ansatz. The ground state is a macroscopic SU(2)symmetric singlet, in which quantum fluctuations suppress longrange order, leading to a spin liquid ground state even in the limit of zero temperature. The elementary excitations are collective spin density fluctuations called spinons, which are fractional excitations carrying spin ½ but no charge. Spinons generated in 1D HAFM through an elementary spinflip process, for example, during inelastic neutron scattering (INS) or resonant inelastic xray scattering (RIXS), are created in pairs. As such, the lowenergy magnetic excitations are spanned by states involving an even number of spinons forming manifolds of two, four, sixspinon … continua and so forth.
The magnetic excitation spectrum has been observed for different realizations of the 1D HAFM by INS^{1,2,5,6,8} and by RIXS^{4,10,20,21,22}. The spectral weight captured by these studies, assigned to the triplet manifold, is located entirely within the boundaries of the twospinon continuum. The reason for this is now well understood through applications of analytical theory^{23,24} or numerical approaches like density matrix renormalization group^{12,13}. While the allowed phase space for fourspinon excitations (and greater) is much larger than for twospinon excitations^{22,24}, kinematic constraints on the matrix elements between the spinon manifolds lead a situation where the multispinon states only contribute significantly for momentum and energy transfers within the boundaries of the twospinon continuum. This picture has been confirmed by detailed comparisons between INS experiments^{5,12} and exact calculations of the dynamical structure factor (DSF)^{5,23,24}, which find the twospinon excitations account only for ~73–74% of the total detected spectral weight, while fourspinon excitations exhaust the majority of the remaining sum rule.
While the exact solution of the pure HAFM model predicts that DSF has a small amount of spectral weight located between the upper boundary of the twospinon continuum and the upper boundary of the fourspinon continuum^{24}, such a small signal has yet to be detected. Fourspinon excitations have been reported outside the twospinon continuum in the metallic 4f electron material Yb_{2}Pb_{2}Pb^{25} and the 1D ferromagnet LiCuVO_{4}^{16,17,18}. Both materials, however, have physics beyond the simple HAFM such as longrange hopping in Yb_{2}Pb_{2}Pb or frustration in LiCuVO_{4}. A direct observation of higherorder spinon excitations separated from the twospinon continuum in the prototypical case of a 1D HAFM with nearestneighbour interactions only is still lacking. Here, we show that RIXS at the O Kedge allows for such an observation, a capability that results from the fundamentally different correlation function that it probes compared to, for example, the spin correlation function of the DSF^{22,26}.
RIXS is a photonin photonout spectroscopy technique where photons inelastically scatter from a sample^{20}. In a RIXS experiment, the photon energy ℏω_{in} of the incident xrays is tuned close to an absorption edge of an atomic species, thereby initiating an electron transition between a core level and an unoccupied valenceband state. This process creates an intermediate state with an additional electron either in the valence or conduction band and a hole in the core level. This corehole excited state will decay on a femtosecond timescale, leaving the system in a longlived valenceband excited state. Since xray photons carry substantial momentum (in contrast to the light of optical or vacuum ultraviolet wavelengths), the triggered valenceband excitations can be studied both in the energy and the momentum domain. Thus, RIXS can be viewed as momentumresolved resonant Raman spectroscopy, suitable for mapping dispersions of excitations in quantum materials. The RIXS selection rules allow studies of magnetic excitations with ΔS_{tot} = 0 (where S_{tot} is the total spin of the system) and—in case of a strong spin–orbit coupling in the initial, intermediate, or final state—ΔS_{tot} = 1.
RIXS has been used to probe electronic excitations involving charge^{4,19,20}, orbital^{4,10,19}, spin^{4,10,14,27,28,29,30}, and lattice^{15,31} degrees of freedom in a wide range of materials. Studies on the dynamic magnetism have largely focused on cuprates, where ΔS_{tot} = 1 direct spinflip excitations can be investigated at the Cu L_{3}edge^{32}. Indeed, in Cu L_{3} RIXS measurements of the quasi1D spinchain cuprate Sr_{2}CuO_{3,} twospinon continuum excitations could be probed (with indications of also fourspinon excitations)^{4}. Studies in other cuprate materials revealed twotriplon excitations in the spinladder system Sr_{14}Cu_{24}O_{41}^{14} and magnon excitations in many quasi twodimensional superconducting cuprates^{27,28,31,32}.
In this article, we report momentumresolved oxygen Kedge RIXS studies of the quasi1D spinchain cuprate Sr_{2}CuO_{3}, one of the best realizations of the 1D HAFM. We observe magnetic excitations in two nonoverlapping regions of phase space. Through detailed modeling within the t−J model, we show that one set of these excitations is quite similar to triplet excitations generally associated with DSF, while the other set corresponds to predominantly fourspinon excitations. Specifically, fourspinon excitations centered at 500 meV energy transfer give a strong and broad response around the Γpoint (q = 0, where q is the momentum transfer along the chain) that is well separated from the boundaries of the twospinon continuum. Our results constitute the discovery of a channel for the creation of magnetic excitations in 1D materials, beyond those resulting from elementary spinflip excitations. We argue that this capability stems from the spin and charge dynamics of the intermediate state, which grants access to fundamentally different correlation functions, not captured by a simple twosite correlation function.
Results
Experimental results
The lowenergy electronic degrees of freedom in the chargetransfer insulator Sr_{2}CuO_{3} are formed from the CuO_{4} plaquettes, which are arranged into 1D cornershared chains^{4,33}, as shown in Fig. 1a. In the atomic limit, the Cu ion is in a d^{9} valence state, with a single hole occupying the Cu 3\(d_{x^2  y^2}\) orbital. There is, however, significant hybridization between the Cu 3d and 2p orbitals of the surrounding oxygen, resulting in a substantial isotropic superexchange interaction J ~ 250 meV^{4,7,8} between the Cu spins. In the real material, the individual CuOCu chains are weakly coupled such that the system has a bulk Néel temperature of T_{N} ~ 5 K^{34}. Above this temperature the chains decouple and become nearly ideal realizations of the 1D HAFM, as evidenced by the observation of the twospinon continuum in INS^{8} and Cu L_{3} RIXS^{4}. The latter RIXS study also found evidence for spin–orbit separation effects in Sr_{2}CuO_{3}, further underscoring the importance of the 1D physics.
Figure 1b shows the xray absorption (XAS) data of Sr_{2}CuO_{3} measured at the O Kedge (a 1s→2p resonance). The intensity reflects the partial density of the unoccupied valence and conduction band states, here projected onto the oxygen orbitals. We observe a sharp excitonic structure in the preedge region and broad continuum states at energies above 529 eV. The excitonic peak corresponds to excitations of the O 1s core electron into the upper Hubbard band (UHB), creating a Cu 3d^{10} state^{33}. This excitation is allowed by the sizable hybridization between the O 2p and Cu 3d orbitals. The UHB XAS peak depends strongly on the polarization of the incident photons reflecting the strong structural and electronic anisotropy of the system^{33}. In particular, the suppression of intensity for σpolarized light indicates that the unoccupied states are oriented in the plane of the CuO_{4} plaquettes, whereas the energy shift upon changing the incidence angle to πpolarized light reflects differences in coordination between the outofchain and the inchain oxygens (indicated in Fig. 1a as sites A and B, respectively), in agreement with previous findings^{35}. For the remainder of this work, we focus on RIXS spectra recorded with the incident photon energies tuned to the UHB B (or A) peak, where an inchain (or outofchain) O 1s core electron is promoted into a neighbouring Cu 3d orbital. This final state of the XAS process dictates the intermediate state of RIXS and is important in determining the scattering crosssection.
Figure 1c shows RIXS spectra measured with the incident photon energy tuned to the resonance of the A and B peak at the O Kedge in comparison with Cu L_{3}edge data at q = π/2a. There are two energy regions with pronounced excitations: one below 1 eV and one above 1.5 eV, separated by a region of very weak spectral weight. The excitations at higher energies are dominated by interorbital dd and charge transfer (CT) excitations^{4,7}; the dd excitations are dominant at the Cu L_{3}edge, whereas the CT excitations are dominant at the O Kedge.
Figure 1d zooms in on the lowenergy excitation region, well below the dd and CT excitations, which is our focus. O K RIXS for photon energies tuned to B with different incident angles are compared to lowenergy Cu L_{3} RIXS data. Below 1 eV we see several excitations. In addition to the elastic line at zero energy transfer, we observe a weakly dispersing excitation at ~90 meV with varying crosssection for the different configurations. This behavior is typical of an optical phonon excitation and the energy scale agrees well with that of a CuO bondstretching lattice vibration^{15}. We, therefore, assign this feature to such a phonon. The line spectrum at q = π/2a (see Fig. 1d, turquoise solid line) reveals a sharp structure coinciding with the very strong spinon excitations at the same qpoint in Cu L_{3}edge data (black line, note that the Cu L_{3} spectrum is divided by a factor 10). In addition, the line spectrum taken close to the Γpoint (red line in Fig. 1d) is dominated by a broad structure, centered at ~0.5 eV and extending to about 1 eV in energy transfer. The energy of this structure is well separated from the dd and CT excitations, suggesting that they are magnetic in origin. A possible path for creating magnetic excitations during RIXS at the O Kedge is sketched in Fig. 1e. This process will be discussed in more detail in the Discussion section.
To probe the dynamic character of the lowenergy magnetic excitations visible in the O Kedge RIXS spectra, we have studied their momentum dependence for momentum transfer along the chain direction, as shown in Fig. 2a. (O Kedge RIXS allows studying about 25% of the first Brillouin zone along [100] towards each side of q = 0, see Supplementary Figure 1.) The experimental geometry is described in Supplementary Note 1 and shown in Supplementary Figure 1. Additional comparisons of the data with the Cu L_{3}edge are provided in Supplementary Notes 2 and 3.
In addition to the strong phonon excitation in O Kedge data, there are two distinct sets of continua in the magnetic region between 0.2 and 1.0 eV. One is dispersing towards zero energy for q = 0 and lies well within the boundaries of the twospinon continuum (indicated by the white dotted lines). The second region is centered at q = 0 and 500 meV energy transfer and is clearly situated outside of the twospinon continuum. The boundaries in Fig. 2 correspond to the twospinon continuum expected for the 1D HAFM model obtained from purely kinematic constraints—see also Supplementary Note 4—assuming a superexchange value of J = 250 meV, as inferred from prior scattering experiments^{4,8}. Comparison to Cu L_{3} data displayed in Fig. 2b, where the twospinon continuum dominates the spectrum, illustrates that O Kedge and Cu L_{3}edge RIXS have quite different responses in terms of the magnetic excitations. However, the line cuts of O Kedge and Cu L_{3}edge RIXS spectra in Fig. 2c show that there is also a finite weight in Cu L_{3}edge RIXS spectra at q = 0. Note that the O Kedge data reveal much stronger polarization dependence due to difference in connectivity of the inchain and outofchain O 2p orbitals (see Supplementary Note 1).
The fact that the Γpoint excitations appear outside the boundaries of the twospinon continuum, and well below the energy transfers where dd and CT excitations occur suggests that they are multispinon in nature. This interpretation is further supported by the fact that they lie completely within the boundaries expected for the fourspinon continuum (indicated by dashed lines, as obtained from pure kinematic arguments—see Supplementary Note 4)^{24}.
Theoretical results
The phase space considerations given above identify the Γpoint excitations belonging to multispinon excitations involving at least four or more spinons. However, to understand the spectral weight of these excitations, it is necessary to compute the RIXS intensity within the Kramers–Heisenberg formalism, due to the prominent role played by the corehole lifetime at the O Kedge. To this end, we performed small cluster exact diagonalization calculations to further elucidate the nature of these excitations. Since we are interested in the energy region well below the dd and CT excitations, we used the t−J model, where these multiorbital processes have been integrated out^{36}. In this case, we adopt values of the hopping integral t = 300 meV and superexchange interaction J = 250 meV, which are consistent with existing literature (see Methods). The computed spectra (with elastic peak removed) are compared against the experimental data in Fig. 2d. Line cuts of the data superimposed over the calculations are shown in Fig. 2e (q ≈ π/2a) and Fig. 2f (q = 0).
The overall agreement between the calculated magnetic response and the experimental data is excellent: our model captures both the dispersing magnetic excitations and the broad continuum centered at the Γpoint. Even the quantitative agreement is very good. Note that our model does not contain the lattice degrees of freedom, and so it does not reproduce the peak associated with the lattice excitations. (If the spin–lattice coupling is weak, we expect that the inclusion of the lattice vibrations would superimpose a phonon excitation on the RIXS spectrum). In this case, the level of agreement between the model and the data in the magnetic region indicates that any spin–lattice coupling is small and that the final states of the O Kedge RIXS process can be well described solely by excitations of the halffilled t−J model, whose final states are the same as those in the Heisenberg model. This observation justifies our neglecting of the lattice excitations and allows us to identify the upward dispersing branch as two and fourspinon excitations, commonly associated with DSF for ΔS_{tot} = 1, while the continuum of excitations centered at q = 0 corresponds to fourspinon excitations that require a more complex correlation function. This assignment is further supported by the dependence of these excitations on the corehole lifetime, which will be discussed shortly.
Discussion
How can we understand the magnetic excitations in RIXS captured by the t−J model, and why do we see magnetic excitations that are absent in INS? In Fig. 3 we illustrate schematically the magnetic excitation mechanisms in a spin chain with the different scattering techniques: INS (Fig. 3a), Cu L_{2,3}edge (Fig. 3b), and O Kedge RIXS (Fig. 3c, d). The ground state of the HAFM is a SU(2)symmetric singlet with S_{tot} = 0. INS measurements connect this ground state to the triplet manifold with S_{tot} = 1 at low temperatures, such that ΔS_{tot} = 1. In a simplified picture (shown here for ΔS_{z} = ±1), such excitation creates two domain walls in the spin chain (Fig. 3a), which decay predominantly into twospinons carrying parallel spins of ½ (due to conservation of angular momentum). Exact calculations^{24} show that these excitations also have overlap with fourspinon excitations, but the majority of the fourspinon weight remains within the boundaries of the twospinon continuum due to kinematic constraints in the matrix elements.
Unlike INS (and RIXS at the Cu L_{2,3}edges), excitations with ΔS_{tot} = 1 are generally forbidden for Kedge RIXS. (This statement holds only for materials with small spin–orbit coupling in the valence band; single flips are allowed in O Kedge RIXS on iridates, see ref. ^{37}.) Instead, ΔS_{tot} = 0 processes like the one sketched in Fig. 1e must be used to create magnetic excitations. Here, the incident photon creates a Cu 3d^{10} UHB excitation in the intermediate state, resulting in a Cu site with an additional electron in direct vicinity to an O 1s core hole (represented in the sketch as blue arrow (spin down) on the lefthand Cu side). The 180° CuOCu bonding angle in Sr_{2}CuO_{3} enables efficient double intersite hopping of 3d electrons between two adjacent Cu sites via the bridging inchain oxygen site (B in Fig. 1a), transferring the Cu 3d^{10} to the neighbouring Cu site (righthand Cu side in Fig. 1e). Since this Cu atom is also hybridized with the oxygen where the core hole is localized, the electron represented by the blue arrow can then decay and fill the core level (see Fig. 1e), leaving the system with a net double intersite spin flip. This process, sketched in Fig. 3c, is analogous to an indirect double spinflip process predicted for Cu K RIXS^{38,39} giving rise to a double domain wall that decays predominantly into two spinons carrying antiparallel spins (due to momentum conservation)^{22}. The crosssection can be related to a dynamic exchange correlation function, whose spectral weight is similar to that of DSF near the zone center^{38}. This excitation pathway explains the presence of the sharper dispersing magnetic excitations in O Kedge RIXS spectra. (Note that in two dimensions a ΔS_{tot} = 0 excitation can only create bimagnon excitations, as each magnon carries a spin of 1). To visualize the scattering process responsible for the creation of fourspinon excitations around the Γpoint the lifetime of the intermediate state plays a critical role.
As we mentioned previously, in Cu L_{2,3}edge RIXS ΔS_{tot} = 1 spinflip excitations that are similar to INS are allowed^{32}. This is possible, since for a Cu 2p corehole the spin–orbit coupling is strong and therefore the change of spin momentum can be compensated by the change of angular momentum. In contrast to INS, however, RIXS involves a doublon in the intermediate state, which decays on a timescale set by the corehole lifetime (~several femtosecond)^{40}. During this time, the additional charge in the intermediate state can interact with the system, creating excitations that are inaccessible through either a single or double spinflip process. For the O 1 s core hole there is no appreciable angular momentum available; therefore, the spin momentum must be conserved and only ΔS_{tot} = 0 excitations are possible (as described above) (Fig. 3c). In a 1D system, the result of this ΔS_{tot} = 0 excitation looks very similar to the result of a single spin flip ΔS_{tot} = 1 in that both excitations lead to the creation of two domain walls, but at the O Kedge they are separated by at least one atomic site and have opposite spins. The lifetime of O 1s corehole states is somewhat longer than the lifetime of Cu 2p corehole states, however. During this time, the doublon in the 3d band can also generate double spinflips on the surrounding sites, as sketched in (Fig. 3d), creating two additional double spinflips separated by larger lattice distances. The subsequent decay of the core hole results in the creation of two additional domain walls, and a total of four spinons in the final state. This scattering channel is the direct result of fluctuations that take place in the intermediate state. Its intensity, therefore, depends on the lifetime of the corehole, as a longerlived doublon will have sufficient time to generate the longerrange double spinflips, separated by large lattice distances. This excitation channel is expected to be weak in Cu L_{3} RIXS, whose corehole is short lived, and completely absent in INS.
We performed calculations for the dependence of these excitations on the lifetime of the intermediate state to test our interpretation. The results are presented in Fig. 4. We observe that upon decreasing the corehole lifetime (increasing Γ), the intensity of magnetic excitations in O Kedge RIXS decreases. Moreover, the spectral weight of the fourspinon excitations moves towards smaller energy transfers (see Fig. 4a). The decrease in intensity is much slower for excitations belonging to the twospinon continuum than for the fourspinon excitations. Whereas the twospinon excitations are still quite pronounced for Γ = 500 meV (Fig. 4b, c), the fourspinon excitations are suppressed below Γ = 300 meV (Fig. 4a), which is comparable to the superexchange interaction J. The suppression of the fourspinon weight at q = 0 proves that the corehole lifetime sets the time scale for the intermediate state to generate these excitations. As its lifetime is quenched below J (~1.3 fs), there is not enough time for additional double spinflips to occur in the chain during the frustrating presence of the doublon. The dynamics of this intermediate state plays an important role for the discovered excitation channel for magnetic excitations and produces additional magnetic correlation functions—beyond a single or double spin flip.
We have demonstrated that RIXS grants access to complementary correlation functions for magnetic scattering compared to INS, which arises from the lifetime and dynamics of the intermediate state. Importantly, this scattering channel is unique to RIXS and provides access to nonlocal spin correlation functions beyond twosite correlation functions probed by traditional scattering techniques. O Kedge RIXS has long corehole lifetimes and is therefore ideal for examining excitations that cannot be detected by INS scattering, as long lifetimes of the intermediate state allow spin and charge fluctuations to take place. We have exploited this fact to observe directly fourspinon excitations of a pure 1D HAFM, located outside the boundaries of the twospinon continuum. This technique opens another avenue to explore quantum magnetism and quasiparticle fractionalization, which has broad applications in the field of quantum magnetism. Timeresolved studies at the upcoming XFEL sources, for example, European XFEL and Swiss FEL, will hopefully facilitate studying such dynamics at the femtosecond timescale.
Methods
Experiment
We applied the technique of highresolution RIXS with the incident photon energy tuned to the O 1s core → 2p UHB resonance (around 528 eV). Singlecrystal samples of Sr_{2}CuO_{3} were grown by the floatingzone method and freshly cleaved before the RIXS experiment. During the experiment the surface normal to the sample, [010], and the propagation direction of the chains, [100], were oriented parallel to the scattering plane. The scattering plane was horizontal. The sample was cooled with a heliumflow cryostat to 14K during the measurements. The experiments were performed at the ADRESS beamline (BL) of the Swiss Light Source at the Paul Scherrer Institut^{41,42}. Incident photons were linearly polarized either in the scattering plane (πpolarization), which was the case for most of the data, or perpendicular to the scattering plane (σpolarization). The XAS data were measured in total fluorescence yield. The BL energy resolution was set to 70 meV or better, with the BL exit slit open to 30 µm. (The BL energy resolution for the Cu L_{3} data^{4} was 100 meV or better, with the BL exit slit open to 10 µm.) The SAXES RIXS spectrometer was located at a fixed scattering angle of Ψ = 130° ± 1°, whereas the incidence angle on the sample varied between 10° ± 1° and 110° ± 1° grazing (see Supplementary Figure 1). The angular horizontal acceptance of the spectrometer was approximately 5 mrad^{40}. The total experimental energy resolution was 80 meV and the simultaneously recorded energy window was 22.2 eV (the total experimental resolution for the Cu L_{3} data^{4} was 140 meV and the simultaneously recorded energy window was 59.2 eV).
Cluster calculations
The RIXS intensity I(q, Ω) was evaluated using the Kramers–Heisenberg formalism where (ℏ = 1)
Here, q = e_{x}·(k_{out}−k_{in}) is the momentum transfer along the xaxis and Ω = ω_{out}−ω_{in} is the energy loss, D is the dipole operator, and \(\left i \right\rangle\), \(\left n \right\rangle\), and \(\left f \right\rangle\) are the initial, intermediate, and final states of the RIXS process with energies E_{i}, E_{n}, and E_{f}, respectively, R_{m} = am is the position of the mth Cu atom, a is the Cu–Cu distance, and Γ is the corehole lifetime. We compute the eigenstates by diagonalizing t−J Hamiltonian defined on a 22 site cluster. The use of this lowenergy effective model is justified since all of the dd and CT excitations appear well above 1 eV in energy loss (see Fig. 1c). Moreover, recent DMRG calculations have explicitly shown that the magnetic excitations probed by Cu Ledge RIXS obtained from a four orbital pd model for Sr_{2}CuO_{3} can be accurately reproduced using an effective t−J Hamiltonian^{36} up to an overall rescaling of the intensity. This result gives us confidence that the downfolded t−J Hamiltonian can capture the magnetic excitations of Sr_{2}CuO_{3}.
The dipole operator in the effective model is given by
where d_{m,σ} annihilates a spin σ hole on Cu site m and \(s_{m,\sigma }^\dagger\) creates a hole in the oxygen 1s orbital on the site between the m and m + 1 Cu sites. Here, the relative phases reflect the phases of the original CuO overlap integrals. The model parameters are t = 300 meV and J = 250 meV, which is appropriate for Sr_{2}CuO_{3}^{4,7}, and Γ = 150 meV for the oxygen Kedge^{15,19}.
Code availability
The source code for the RIXS calculations is available from the authors upon reasonable requests. Requests for the code should be directed to S.J.
Data availability
The data that support the findings of this study are available from the authors upon reasonable requests. Requests for experimental data should be directed to J.S. and T.S.
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Acknowledgements
We thank C. Batista and K. Wohlfeld for useful discussions. The experiments were performed at the ADRESS BL of the Swiss Light Source at the Paul Scherrer Institut. We acknowledge support from the Swiss National Science Foundation and its NCCR MaNEP. CPU time was provided in part by resources supported by the University of Tennessee and Oak Ridge National Laboratory Joint Institute for Computational Sciences (http://www.jics.utk.edu).
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J.S. and T.S. designed the experiment. J.S., T.S., and K.J.Z. performed the experiment with the assistance of V.N.S. M.M., H.M.R., and L.P. contributed to the discussion of data. J.S. performed data analysis in discussion with T.S., U.K., and S. J. performed theory calculations. S.S. and A.R. have prepared and characterized the crystal samples. T.S. and S.J. were responsible for project management. J.S. wrote the paper together with U.K., S.J., and T.S. with input from all other authors.
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Schlappa, J., Kumar, U., Zhou, K.J. et al. Probing multispinon excitations outside of the twospinon continuum in the antiferromagnetic spin chain cuprate Sr_{2}CuO_{3}. Nat Commun 9, 5394 (2018). https://doi.org/10.1038/s4146701807838y
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DOI: https://doi.org/10.1038/s4146701807838y
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