Multi-spinon and holon excitations probed by resonant inelastic x-ray scattering on doped one-dimensional antiferromagnets

Resonant inelastic x-ray scattering (RIXS) at the oxygen $K$-edge has recently accessed multi-spinon excitations in the one-dimensional antiferromagnet (1D-AFM) \sco, where four-spinon excitations are resolved separately from the two-spinon continuum. This technique, therefore, provides new opportunities to study fractionalized quasiparticle excitations in doped 1D-AFMs. To this end, we carried out exact diagonalization studies of the doped $t$-$J$ model and provided predictions for oxygen $K$-edge RIXS experiments on doped 1D-AFMs. We show that the RIXS spectra are rich, containing distinct two- and four-spinon excitations, dispersive (anti)holon excitations, and combinations thereof. Our results highlight how RIXS complements inelastic neutron scattering experiments by accessing additional charge and spin components of fractionalized quasiparticles.

Introduction -One-dimensional (1D) magnetic systems have attracted considerable interest throughout the scientific community for more than half a century. This interest stems from the fact that these systems provide excellent opportunities to study novel quantum phenomena such as quasiparticle fractionalization or quantum criticality. Moreover, model Hamiltonians of 1D systems can often be solved exactly using analytical or numerical techniques, making them ideal starting points for understanding the physics of strongly correlated materials. For example, the exact solution of the 1D Hubbard model by Lieb and Wu [1] represented a breakthrough in the field, showing that interacting electrons confined to 1D are characterized by spin-charge separation, where electronic quasiparticle excitations break into collective density fluctuations carrying either spinless charge ("holons") or chargeless spin ("spinons") quantum numbers with different characteristic energy scales. This work inspired an intense search for materials showing spin-charge separation, but it has only been in the last two decades that this phenomenon was observed [2][3][4][5][6][7].
Resonant inelastic x-ray scattering (RIXS) [8] has evolved as an important tool for studying the magnetic excitations in correlated materials [9][10][11], complementing inelastic neutron scattering (INS). RIXS, however, is also a powerful probe of orbital and charge excitations, as was succinctly demonstrated by the experimental observation of spin-orbital fractionalization in a Cu L-edge RIXS study of Sr 2 CuO 3 [12,13]. Sr 2 CuO 3 contains 1D chains of corner-shared CuO 4 plaquettes, where a single hole occupies each Cu 3d x 2 −y 2 orbital, forming a quasi-1D spin-1 2 chain. Due to a very weak interchain interaction, the CuO 3 chains decouple above the bulk ordering temperature T N = 5.5 K and form a nearly ideal realization of a 1D antiferromagnet (AFM) [14]. A recent O Kedge RIXS study [15] of undoped Sr 2 CuO 3 directly observed multi-spinon excitations outside of the two-spinon (2S) continuum (see also Fig. 1) further highlighting the potential for RIXS to probe such excitations.
To date, spin-charge separation has not been observed using RIXS [13]. In this letter, we performed exact diagonalization (ED) and DMRG calculations to show that RIXS measurements on doped 1D AFMs can fill this need. Specifically, we show that O K-edge RIXS can access multi-spinon excitations, antiholon excitations, and combinations thereof, thus providing a unique view of spin-charge separation in doped 1D AFMs. Since Sr 2 CuO 3 can be doped with Zn, Ni, or Co [16,17], this material can be used to test our predictions.
Magnetic Scattering at the O K-edge -Before proceeding, we review how magnetic excitations occur in the O K-edge (1s → 2p) [18] measurements on Sr 2 CuO 3 , as sketched in Fig. 1(a). Sr 2 CuO 3 is a charge-transfer insulator and the ground state character of the CuO 4 plaquettes is predominantly of the form α|d 9 + β|d 10 L (α 2 ≈ 0.64, β 2 ≈ 0.36) [19,20], due to hybridization between the Cu 3d x 2 −y 2 and O 2p orbitals. Here, L denotes a hole on the ligand O orbitals. Due to this hybridization, the incident photon can excite an O 1s core electron into the Cu 3d orbital when tuned to the O K-edge, creating an upper Hubbard band excitation. In the intermediate state, the d 10 configuration can move to the neighboring Cu ion via the bridging O orbital. Since the adjacent Cu orbital also hybridizes with the O containing the core hole, one of the d 10 electrons can then decay to fill it, creating a final state with a double spin flip.
The dynamics in the intermediate state are essential for generating magnetic excitations at this edge, and this is a fundamental difference in how RIXS and INS probe magnetic excitations. One of the advantages of working at the O K-edge is that it has relatively long core-hole lifetimes ( /Γ, Γ = 0.15 eV [21]) in comparison to other edges (Γ = 1.5 eV at the Cu K-edge and 0.3 eV at the Cu L 3 -edge [22]), which provides a longer window for generating magnetic excitations [15,23]  necessary. Several efforts addressing the spin dynamics in RIXS have mostly used the ultrashort core-hole lifetime (UCL) approximations, which applies to edges with short core-hole lifetimes [24,25], while studies of 1D systems beyond UCL approximations have been limited [22,26]. Ref. 26 studied the effect of incidence energy on spin dynamics RIXS spectra in 1D using small cluster ED, but a systematic analysis of the incident energy dependence was not carried out. As a result, the multi-spinon excitations at q = 0 were not reported. Similarly, Ref. 27 discussed the doping dependence of the RIXS spectrum for the t-J model by evaluating the spin response, but the charge response along with the intermediate state dynamics were left out. For these reasons, these prior studies could not address the physics reported here.
Model and methods -We model the RIXS spectra of Sr 2 CuO 3 using a 1D t-J Hamiltonian Here,d i,σ is the annihilation operator for a hole with spin σ at site i, under the constraint of no double occupancy, n i = σ n i,σ is the number operator, and S i is the spin operator at site i. During the RIXS process [8], an incident photon with momentum k in and energy ω in ( = 1) tuned to an elemental absorption edge resonantly excites a core electron into an unoccupied state in the sample. The resulting core hole and excited electron interact with the system creating several elementary excitations before an electron radiatively decays into the core level, emitting a photon with energy ω out and momentum k out . The RIXS intensity is given by the Kramers-Heisenberg formula [8] (1) where Ω = ω in − ω out is the energy loss, |i , |n , and |f are the initial, intermediate, and final states of the RIXS process with energies E i , E n , and E f , respectively, and D is the dipole operator for the O 1s → 2p transition. In the downfolded t-J model D takes the effective form where q (= k out −k in ) is the momentum transfer and the relative sign is due to the phases of the Cu 3d x 2 −y 2 and O 2p x orbital overlaps along the chain direction. Here, s i+ 1 2 ,σ is the hole annihilation operator for the 1s core level on the O atom bridging the i and i + 1 Cu sites.
In the real material, the core hole potential raises the on-site energy of the bridging oxygen orbital (in hole language) in the intermediate state while exerting a minimal influence on the Cu sites. This change locally modifies the superexchange interaction between the neighboring Cu atoms [28]. To account for this effect, we reduce the value of J i,i+1 = J/2 when solving for the intermediate states, where the core-hole is created on the O atom bridging the i and i+1 sites. Our results are not sensitive to reasonable changes in this value [29].
Throughout we set t = 1 as our unit of energy (t ≈ 300 meV in Sr 2 CuO 3 ). The remaining parameters are Γ n = 1 2 t for all n and J = 5 6 t, unless otherwise stated. These values are typical for the O K-edge measurements of Sr 2 CuO 3 [12,20,21,30]. We also introduce a Gaussian broadening (Γ = 1 3 t) for energy conserving δ-function appearing in Eq. (1). We evaluated Eq. (1) on a L = 20 site chain using the Lanczos method with a fixed filling.
To help identify the relevant charge and spin excitations in the RIXS spectra, we also performed DMRG simulations [31,32] for the dynamical charge N (q, ω) and spin S(q, ω) structure factors on an L = 80 site chain, and using correction-vector method [33,34]. Within the correction vector approach, we used the Krylov decomposition [35] instead of the conjugate gradient. In the ground state and dynamic DMRG simulations, we used a maximum of m = 1000 states, keeping the truncation error below 10 −6 and used a broadening of the correction-vector calculation of η = 0.08t [29]. The computer package dmrg++ developed by G. Alvarez,

FIG. 2.
Calculated RIXS spectra at ωin = 5t for a L = 20 site doped t-J chain with (a) one and (b) two additional doped electrons. The white, black, and red lines shows the boundaries for two-spinon continuum, the dispersion of the antiholonic excitation, and upper boundary for antiholon-2S excitations, respectively. Panels (c), (d), and (e) compare the doped RIXS spectra with the undoped case for momentum transfers of q = π/a, q = π/(2a), and q = 0, respectively.

CNMS, ORNL, was used in the DMRG simulations [36].
Undoped RIXS spectra - Figure 1(b) shows the RIXS intensity for the half-filled t-J chain, reproduced from Ref. 15. For comparison, Fig. 1(c) shows S(q, ω) obtained using DMRG for the same parameters. The RIXS intensity has two main features. The first is a continuum of excitations that closely mirrors S(q, ω) and is situated within the boundaries of the 2S continuum. Its intensity is relatively independent of the incident photon energy and is associated primarily with 2S excitations [15,37]. The second feature is a continuum of excitations laying outside of the 2S continuum, corresponding to 4S excitations. Its intensity is sensitive to both the incident photon energy and the core-hole lifetime, indicating that the intermediate state plays a critical role in creating those excitations [15].
Doped RIXS spectra -We now turn our attention to the results for the doped case. Figures 2(a) and 2(b) show the RIXS intensity obtained on a 20-site chain at 5% and 10% electron doping, respectively. Here, we have used ω in = 5t to enhance the intensity of the features appearing at q = 0. To help us better understand the main features, we also computed S(q, ω) ( Fig. 3(a)) and N (q, ω) ( Fig. 3(b)) for 5% doping using DMRG.
The RIXS spectra for the doped cases have three recognizable sets of features: i) a continuum that mirrors the S(q, ω) in Fig. 3(a); ii) a cosine-like dispersive feature with a bandwidth of 4t that mirrors N (q, ω) in Fig. 3(b); and iii) two continua, centered at q = 0 and extending up to ∼ 6t in energy loss. These features are absent in S(q, ω) and N (q, ω). The excitations (i) and (ii) point to a manifestation of spin-charge separation in that the response bifurcates into primarily two-spinon (i) and antiholon (ii) excitations, characterized by different energy scales. Also, notice that the dispersions of various peaks in Figs. 2(a) and 2(b) do not vary significantly with a small change in doping, except for their relative intensities.
Figures 2(c)-2(e) compare the doping evolution of the RIXS features at fixed momentum points. Fig. 2(c) shows q = π/a, where the upper bound (πJ) of the spin excitations decreases upon doping. Similarly, the linecut at q = π/2a in 2(d) shows that the lower bound (πJ/2) of the 2S continuum also decreases with doping, allowing for final states below the 2S continuum of the undoped case. We also observe a secondary feature at higher energy loss due to changes in the holon branch and 4S excitations. Fig.2(e) shows a cut at q = 0, where two distinct sets of peaks are clear. The group at lower energy losses appears in the same energy range of the multi-spinon peak observed in the undoped case. The peaks at higher energy loss appear above Ω = 4t and are identified below.
The calculated spectra can be understood by making use of the spin-charge separation picture: in 1D, the wavefunction of the large U Hubbard model for N electrons in L lattice sites is a product of 'spinless' charge and 'chargeless' spin wavefunctions. [38][39][40] The dispersion of charge excitations is given by ωh(kh) = −2t cos(kha) [41,42], which agrees well with the dispersion observed for feature (ii) (see black dashed line) and in N (q, ω). As shown in the supplemental material [29], the N (q, ω) computed here for small electron doping is identical to the N (q, ω) obtained for a 1D spinless fermions chain with the same fermionic filling, supporting the spin-charge separation picture. This result indicates that the charge excitation is behaving like a nearly free spinless quasiparticle, i.e. a holon/antiholon. Concerning the spin part, the dispersion relation for a single spinon is given by ω s (k s ) = π 2 J| sin(k s a)|. Due to the RIXS selection rules, these spin excitations must be generated in even numbers, resulting in a continuum whose boundaries are defined by this dispersion relation. At small doping, the limits of this continuum are modified, which is accounted for using a slightly modified su-perexchangeJ = J n [40]. The upper and lower boundaries of the modified 2S continuum are indicated by the white lines in Fig. 2 and agree well with the observed excitations.
We can summarize the picture emerging from our results as follows: the 2S-like continuum present in the RIXS spectrum is a pure magnetic excitation as it compares well with the S(q, ω) from DMRG. The dispersing cosine-like feature in the doped RIXS spectra compares well with the N (q, ω) from DMRG. We have verified that the N (q, ω) of the spinless fermions with occupations equal to the electron-doping considered above are qualitatively similar to the results obtained for the doped t-J chain [29]. We therefore assign this feature to purely charge-like antiholon excitations.
The peaks at q = 0 of the RIXS spectrum are not captured by either S(q, ω) or N (q, ω). The lower continuum resembles the multi-spinon continuum [15] also observed in the undoped case, and we, therefore, associate it with 4S excitations. Conversely, the continuum of excitations at energy losses between 4t and 6.5t (well beyond the upper boundary of 4S continuum [2πJ (= 5.24t)] [43]) is unique to the doped case. The excitations are bounded by 4t + πJ cos(q/2) (dotted red line), which one obtains from a simple convolution of the antiholon and twospinon excitations. Therefore, we assign these to an antiholon plus two-spinon final state. The fact that the intensity and distribution of these excitations are very sensitive to doping supports this view. As we further increase the doping, we see additional spectral weight above the 4t + πJ cos(q/2) boundary, indicating that these quasiparticle interactions are beginning to interact to produce modified dispersion relationships.
Incidence energy dependence - Figure 4 shows the changes in the RIXS intensity maps as the incident photon energy is varied from ω in = 3t to 8t for the 5% doped case. (The results at 10% doping are similar and provided in Ref. 29.) The final state excitations resembling S(q, ω) and N (q, ω) are clear in all cases, but there are some variations in the overall intensity as ω in is tuned through the XAS resonance peak (Fig. 4a, inset). The remaining excitations exhibit a strong incident energy dependence, where both antiholon excitations and the multi-spinon/antiholon excitations centered at q = 0 are difficult to resolve for ω in / ∈ (3t, 8t). By varying ω in , one selects particular intermediate states |n in the RIXS process. The incident energy dependence shown in Fig. 4 indicates that only certain intermediate states can reach the multi-particle excitations centered at q = 0.
Discussion -Several previous theoretical works have calculated the RIXS spectra for 1D t-J [24,25] and Hubbard [22,26] chains using the same formalism. In the doped and undoped cases, these studies obtained RIXS spectra resembling S(q, ω); however, they did not capture the (anti)holon or multi-spinon excitations observed here. Refs. [24,26], obtained nonzero weight in the q = 0 response but with a significantly reduced spectral weight in comparison to our results. In RIXS at oxygen K-edge, only ∆S = 0 excitations are allowed. Ref. 22 and 26 showed that ∆S = 0 excitations vanishes at q = π/a, whereas we have the maximum at that point in our model. We believe that this discrepancy is due to the lack of hopping from the core-hole site due to the strong corehole potential used in that work, which is appropriate for the Cu L and K-edges. A strong core-hole potential will tend to localize the excited electrons in the intermediate state, thus suppressing its dynamics. We can confirm this in our model by setting t = 0 in the intermediate state for the undoped system, which also prohibits charge fluctuations and produces spectra similar to Refs. 22 and 26. Furthermore, given the sensitivity to ω in shown in Fig.  4, prior studies may have missed the relevant excitations due to their choice of incident energies.
In summary, we have shown that spin-charge separation can be observed in O K-edge RIXS on doped 1D-AFMs and that these systems exhibit remarkably rich spectra consisting of multi-spinon and holon excitations. Our results highlight the potential for RIXS to simultaneously access the charge, spin, and orbital degrees of freedom in fractionalized quasiparticle excitations, applicable to many quantum materials. Department of Energy, Office of Basic Energy Sciences, Materials Sciences and Engineering Division. This research used computational resources supported by the University of Tennessee and Oak Ridge National Laboratorys Joint Institute for Computational Sciences.