Abstract
The nature of the spin excitations in superconducting cuprates is a key question toward a unified understanding of the cuprate physics from longrange antiferromagnetism to superconductivity. The intense spin excitations up to the overdoped regime revealed by resonant inelastic Xray scattering bring new insights as well as questions like how to understand their persistence or their relation to the collective excitations in ordered magnets (magnons). Here, we study the evolution of the spin excitations upon holedoping the superconducting cuprate Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} by disentangling the spin from the charge excitations in the experimental cross section. We compare our experimental results against density matrix renormalization group calculations for a tJlike model on a square lattice. Our results unambiguously confirm the persistence of the spin excitations, which are closely connected to the persistence of shortrange magnetic correlations up to high doping. This suggests that the spin excitations in holedoped cuprates are related to magnons—albeit shortranged.
Introduction
Starting from antiferromagnetic Mott insulators, the cuprate hightemperature superconductors go through various quantum states with the charge carrier doping as the tuning parameter and form a universal dopingtemperature phase diagram. For the holedoped cuprates, superconductivity emerges in the intermediate doping regime in close proximity to the notorious pseudogap state and the recently established charge density wave state^{1}. Disentangling the physics behind these intertwined states is a major challenge for constructing a complete theory of the superconducting cuprates. Fundamentally, the competition between the exchange energy of the localized spins and the kinetic energy of the doped holes is believed to dominate the basic physics, and the two energy scales are naturally regulated by the amount of doped holes^{2}. While the holes tend to delocalize and turn the system into a Fermi liquid at high doping level, electron correlations and local antiferromagnetic correlations survive with modest doping in the superconducting regime and coexist with the welldefined quasiparticles^{2,3,4,5}. Exactly how these spin and charge degrees of freedom act and interact throughout the dopingtemperature phase diagram is therefore a crucial question towards the formulation of a definitive theory of superconductivity in the doped cuprates.
Experimentally, the dynamics and interactions of the spin and charge degrees of freedom are studied by assessing the momentum and energy dependence of the elementary excitations across the phase diagram. As suggested in numerous papers over the last 11 years^{6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27}, resonant inelastic Xray scattering (RIXS) observes intense spin excitations in a large momentum space area around the Brillouin zone center, which persist well into the overdoped region. Crucially the spin excitations in doped samples disperse along the (π, 0) direction similarly to the magnons in the antiferromagnetic phase with the damping increasing moderately, although they rapidly become overdamped with doping along the (π, π) direction and show almost nondispersive profiles^{14,15,17,20,21,22,23,26,27}. This suggests that in some particular area of the Brillouin zone, that is mostly in the (π, 0) direction, the magnetic excitations completely ‘ignore’ the existence of a critical value of the doping δ at which Fermi liquid behavior takes over the correlated magnetism and, moreover, that these excitations resemble the wellknown magnons in undoped cuprates (hence their name—paramagnons). Naturally, such results are very much counterintuitive for they lead to an apparent paradox related to the small changes of the spin excitations in the doped cuprates, despite the rapid collapse of the longrange magnetic order upon doping and the dominant Fermiliquid nature in the overdoped regime. This has sparked an intensive discussion on whether the observed magnetic excitations should indeed be viewed as paramagnons–or rather incoherent particlehole excitations with a spinflip^{14,15,24,28,29,30}.
To justify the nature of the spin excitations in cuprates as well as the reason of their persistence upon doping, it is necessary to precisely evaluate the momentum and doping evolution of the intrinsic spin excitations and comprehensively compare to theoretical calculations. This is a difficult task due to the experimental difficulties in extracting S(q, ω) from RIXS spectra and also due to the problems in reliably calculating S(q, ω). One of the major experimental obstacles comes from the mixing of spin and charge excitations in the RIXS spectra of doped cuprates^{31,32}. With increasing doped holes, one would expect the charge excitations to be stronger, which will worsen the ‘mixing’ problem. This strongly hampers the correct assignment of the spectral profile to solely spinflip containing excitations, and casts doubts on whether RIXS indeed observes the persistence of the intrinsic spin excitations.
Here we report a systematic study on the momentum and doping evolution of the disentangled intrinsic spin and charge excitations in superconducting Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} (Bi2212). By applying the azimuthaldependent analysis based on the distinct scattering tensors^{33}, we show that the lowenergy excitations of the Cu L_{3}edge RIXS spectra can be well described by spinflip (spin) and nonspinflip (charge) components for all studied doping levels and momenta, which allows us to extract the intrinsic spectral weights of the two components. We find that the spin spectral weight only slightly increases (decreases) with doping at intermediate momentum q along the (π, π) [(π, 0)] direction, which unequivocally confirms the persistence of the spin excitations in doped cuprates. We then compare the above experimental results to stateoftheart density matrix renormalization group (DMRG) calculations of tJlike models. The detailed comparison reveals the key characteristics of the spin excitations in the doped cuprates: On one hand, we unravel the crucial role of the longerrange hoppings, as the second and thirdnearestneighbor hopping are needed to fully reproduce the experimental spin spectrum; on the other hand, we show that solely the shortrange magnetic correlations are enough to qualitatively reproduce the persistence of the intensity of the spin excitations upon doping. Our results thus suggest that RIXS indeed observes the persistent spin excitations in holedoped cuprates and that their paramagnetic nature can be understood as stemming from localized spins with shortrange correlations.
Results
Disentangling the spin and charge excitations in the RIXS spectra
We studied the spin and charge excitations of three different doped Bi2212 samples from underdoped to overdoped regime. The three samples are labeled as UD (T_{c} = 73 K), OD1 (T_{c} = 88 K) and OD2 (T_{c} = 65 K), as shown in Fig. 1a. To disentangle the overlapping spin and charge excitations in the Cu L_{3}edge RIXS spectra, we applied the recently proposed azimuthaldependent method^{33}, which resolves the two kinds of excitations based on their distinct scattering tensors. In this method, a sample on a wedged sample holder is rotated to change the orientation of the photon polarization in the sample space (see Fig. 1b and Methods section), which gives rise to different rotation dependences according to the scattering tensors^{33}. Fig. 1c, d show the azimuthal dependences of the nonspin flip and spinflip excitations at σ and π incident polarizations with a 40^{∘} wedge angle. In the RIXS experiment, selfabsorption effects need to be considered, which depends on the scattering geometry (azimuthal angles in this experiment) as well as the polarization and energy of the photons. Fig. 1e, f show the azimuthal dependence of spin and charge response after including the selfabsorption effect at an energy loss (E_{f} − E_{i}) of −0.25 eV (see Supplementary Note 1 for further details). The spin and charge responses show clearly different azimuthal dependences, which allows to disentangle them in the lowenergy RIXS spectra.
Figure 2 a and b show the RIXS intensity map of OD2 sample at q = (0.33, 0) as a function of azimuthal angle ϕ and energy loss E at σ and π incident polarization, respectively. As the energies of dd excitations are situated well above 1 eV^{14,15,34}, it is natural to assume that the lowenergy excitations are mainly composed of the singlespinflip and nonspinflip excitations involving the 3\({d}_{{x}^{2}{y}^{2}}\) orbital. Although the double spin flip with a net spin change of zero (bimagnon with ΔS = 0) is also allowed in RIXS process in cuprates^{25,35,36}, earlier studies show that its spectral weight quickly diminishes with holedoping^{37,38,39}, and becomes negligible in Cu L_{3} RIXS when doping is beyond 0.08^{25}. In addition, the bimagnon intensity in Cu L_{3} RIXS is maximal at the zone center, and becomes significantly weaker at large momentum^{25}. The lowenergy RIXS intensity at a certain momentum q is thus expressed as a linear combination of the spectral weights of the singlespinflip and nonspinflip components modified by their corresponding azimuthal dependence:
where w_{s(c)}(E) is the spectral weight of spin (charge) components, and A_{s(c)}(E, ϕ, ϵ_{i}) is the azimuthal dependence which is already known from the scattering tensor and selfabsorption effects, and ϵ_{i} is the incident polarization. Fig. 2cf show the ϕ dependence of RIXS intensity at constantenergy loss of 0 and −0.25 eV with the decomposed spin and charge contributions w_{s(c)} ⋅ A_{s(c)}(ϕ). As can be seen, the RIXS azimuthal dependences can be well fitted by these two components, which verifies that the above analysis correctly describes the lowenergy RIXS response in Bi2212. In addition, the quasielastic scattering at 0 energy loss is dominated by chargelike ϕ dependence, consistent with the charge nature of the quasielastic peak. In Fig. 2g, h, we compare the RIXS spectra at two special geometries, grazing incidence (ϕ = 0^{∘}) with σ polarization and grazing emission (ϕ = 180^{∘}) with π polarization, with the decomposed spin and charge components. The grazing emission with π polarization is usually used to measure the magnetic excitations in cuprates, since the charge component is largely suppressed in this geometry as shown in Fig. 2h. Nonetheless, the charge component is still considerable, which could influence the correct evaluation of the profile and intensity of magnetic excitations. It is therefore necessary to fully disentangle the spin and charge components to precisely study their nature. We note here that the obtained spectral functions w_{s(c)}(E) are solely related to the properties of the studied samples, as the angle and polarization related geometry factors and the selfabsorption effect are removed by the knowledge of A_{s(c)}(E, ϕ, ϵ_{i}). This allows the direct and unambiguous comparison between w_{s(c)}(E) and theoretical calculations based on different models, which could provide vital knowledge to understand the spin and charge excitations in cuprates.
Figure 3 presents the obtained spin and charge spectral functions w_{s(c)}(E) of the three different doped samples at different momenta q. Fig. 3a–g show the decomposed spin spectral functions, which all show a single peak with a damped profile. Error bars describe the fitting errors in the disentanglement (see Methods). Fig. 3h plots the energy integrated intensity I_{s}(q) of the spin spectral functions. The I_{s}(q) of all three samples show a similar q dependence: I_{s}(q) monotonically increases with increasing q, and has a slightly larger intensity when approaching large q along (π, π) direction than (π, 0) direction. This q dependence is qualitatively consistent with the results in a previous study which calibrate the geometry influences by comparing to the INS results^{27}. Fig. 3i–o show the decomposed charge response at different q. There are two main components in the charge response: a quasielastic peak including the lowenergy phonons close to zeroenergy loss, and a broad peak around −0.4 eV which extends to highenergy loss. The quasielastic peak is enhanced at (0.13, 0.13), which is due to the structure modulation at (0.125, 0.125) in Bi2212 samples. Fig. 3p shows the integrated intensity of the broad peak after the quasielastic peak is subtracted. In contrast to the spin response, this charge response shows much stronger intensity increases along (π, 0) direction while it saturates around (0.25, 0.25) along (π, π) direction. The difference further highlights the distinct nature of the two decomposed components.
By comparing different doping levels, one can notice that the spin excitations show different development along the (π, 0) and (π, π) directions: the total spectral weight increases with increasing doping at intermediate q along (π, π), while it slightly decreases along (π, 0) direction, as shown by I_{s}(q) in Fig. 3h. On the other hand, the spectral weights of the decomposed charge excitations simply increases with increasing doping along both directions, while the increase is more remarkable along (π, 0) direction than (π, π) as shown in Fig. 3p. A previous RIXS study on single layer (Bi,Pb)_{2}(Sr,La)_{2}CuO_{6+δ}^{26} also investigated the influence of doping on the spin excitations using the grazingemission and incident πpolarization geometry which enhances the scattering contribution from the spinflip channel. In contrast, they found the intensity of spin excitations increases with doping at small and intermediate q along both (π, 0) and (π, π) directions, but crosses over to decrease at large q. These different results could originate from either the differences between single and double layer cuprates, or a small residual mixture with charge excitations in the grazingemission and incident πpolarization geometry. Note that the (π, 0) direction will endure more influences from the residual charge excitations as the charge excitations are more intense and dopingdependent along the (π, 0) direction (see Fig. 3p). The distinct spin excitations response to the hole doping along (π, 0) and (π, π) directions could provide important information for the understanding of the spin dynamics in doped cuprates, which can be obtained by comparing with stateoftheart theoretical calculations discussed in detail in the next part of the paper.
For a more quantitative analysis of the spin response, we fit the spin spectral functions w_{s(c)}(E) by a damped harmonic oscillator (DHO) model convoluted with a resolution function (see Methods). As shown by the solid lines in Fig. 3a–g, the results can be well fitted by the DHO model. Fig. 4 presents the fitting results. The bare frequency ω_{0} is similar for all three dopings, while the damping γ increases with increasing doping and has a much larger value along (π, π) direction. This is consistent with previous studies on doped cuprates showing that the magnetic excitations are much more damped along (π, π) direction^{14,15,17,20,21,22,23,26,27}. With the charge excitations excluded, we can now rule out that the overdamped profile of the spin excitations along (π, π) direction comes from an increase of charge contributions with doping. We note that the fitted damping factors of the two smallest momenta of OD2 sample are large and out of the main trend of the momentum dependence. We attribute this to the decomposed shape of spin excitations bearing more influences from the uncertainties in the azimuthal fitting when the spin excitation peak is getting closer to the elastic peak at small momenta. Fig. 4b shows the propagation frequency defined as \({\omega }_{{{{\rm{p}}}}}=\sqrt{{\omega }_{0}^{2}{\gamma }^{2}}\), where a zero value is assigned when the system is over damped, i.e. ω_{0} < γ. One can see that the spin excitations in the overdoped OD2 sample are fully over damped along the (π, π) direction. Fig. 4c shows the fitted proportional amplitude A to DHO model. It increases with increasing doping along (π, π) direction while it changes little along (π, 0) direction, which is a bit different from the integrated total spectral weight I_{s}(q) shown in Fig. 3h. This is mostly due to I_{s}(q) including both the effects from the proportional amplitude A and the damping γ, while A excludes the effect of damping γ which suppresses the total intensity.
Unraveling the character of the spin excitations in doped cuprates
The model of choice to study the evolution of the spin excitations upon doping the cuprates is the “celebrated” tJlike model^{40}, defined by the Hamiltonian on a 2D square lattice:
where \({\tilde{c}}_{i}^{{\dagger} }\) operator creates an electron at site i in the constrained Hilbert space without double electron occupancies, S_{i} is a spin1/2 operator at site i and \({\tilde{n}}_{i}\) is the onsite electron density at site i: \({\tilde{n}}_{i}={\tilde{c}}_{i}^{{\dagger} }{\tilde{c}}_{i}\). The model parameters t, \({t}^{{\prime} }\) and \(t^{\prime \prime}\) denote the hopping integrals between first, second, and third neighbors, respectively, whereas J is the antiferromagnetic Heisenberg interaction between nearest neighbor spins. In our calculations, we take a realistic and widelyaccepted (cf.^{41} or a quite similar choice in ^{42}) choice of the values of the t\({t}^{{\prime} }\)\(t^{\prime \prime}\)J model parameters \({t}^{{\prime} }=0.3t\), \(t^{\prime \prime}\) = 0.15t and J = 0.4t. These values slightly differ from the dopingdependent values suggested for Bi2212 in^{43}, but we keep these more standard values to make our study universal. Furthermore, to better understand the role of the longerrange hoppings, we consider switching off the \(t^{\prime \prime}\) hopping (the t\({t}^{{\prime} }\)J model), both the \(t^{\prime \prime}\) and \({t}^{{\prime} }\) hoppings (the tJ model) as well as substantially increasing the value of \({t}^{{\prime} }\) in the t\({t}^{{\prime} }\)J model calculations. Finally, note that, while to describe electronic properties of the cuprates probably the chargetransfer (pd) model would be more appropriate, the spin excitations are believed to be welldescribed by models with oxygens being integrated out^{26,31,44,45,46,47,48,49}. The t(\({t}^{{\prime} }\)\(t^{\prime \prime}\))J model on a square lattice has been intensively studied as a host of the superconducting state for a long time (for example, see^{50,51} and references therein), and a possible existence of the superconducting phase in a wide range of hole doping has been proposed^{52}.
Next our goal here is to compute the static spin structure factor S(q), typically defined as:
where the i, j indices run over all sites, N is the number of sites and r_{i} denotes the position of the site in the lattice. This is done by involving stateoftheart DMRG calculations which are carried out on an N = 6 × 6 square lattice with open boundary conditions (OBC). We chose a 6x6 OBC cluster to investigate a wide range of parameters with high accuracy. The use of OBC enables us to avoid an artificial enhancement of specific period correlations, which frequently occurs in periodic and cylindrical systems. However, due to the OBC, the charges tend to localize at the edges when charge imbalance, i.e. holes, is introduced. To counterbalance this effect, we introduce an edge factor λ that multiplies the electronic hopping parameters t, \({t}^{{\prime} }\) and \(t^{\prime \prime}\) as well as the spin exchange coupling J acting on the sites on the perimeter of our cluster, see Methods and Supplementary Note 6.
By using the real space approach we have full control over which contributions to S(q) we include in Equation (3). In fact, we can select the distance ℓ at which the sum in Equation (3) is truncated. In particular, it is possible to consider the different total averages for the different spinspin correlations 〈S⋅S〉_{ℓ} with ℓ = 1, 2, 3, … defining the considered neighbors. The difference in S(q) between using the singular values of the spinspin correlations and the averages is minimal (see Supplementary Note 7). While shielding us from accessing S(q, ω), this approach allows us to thoroughly study the possible magnonic character of the persistent spin excitations.
We now discuss how our theoretical calculations compare to the experimental results presented in Fig. 3. Our aim within the calculations is to reproduce the switching in the sequence of intensities upon doping. Hence, our focus is on this qualitative aspect of the experimental results, rather than on the quantitative reproduction of the experimental data within our theoretical calculations. The main results are shown in Fig. 5: whereas Fig. 5a presents the experimental integrated intensity of spin spectral weights for the three measured doping levels UD, OD1, and OD2 (see above), Fig. 5b shows the calculated S(q) with the t\({t}^{\prime}\)\(t^{\prime \prime}\)J model in the restricted Brillouin zone kinematically available to the experiments. Finally, we schematically compare the doping evolution of the experimental and theoretical intensities at all experimental momenta in Fig. 5c. The crucial message here is that, overall, there is good qualitative agreement between theory and experiment. First, the experimentally observed small anisotropy between the (π, π) and (π, 0) directions—the (π, π) direction shows larger intensities than (π, 0) at high q—is also reproduced by our calculations, although it is not as small as in the RIXS experiment. Second, at five crucial momentum points the theoretical calculations give the same sequence of intensities of the spin structure factor as a function of doping as the face value of the experimental results (see five gray circles in Figs. 5b and 5c). In addition, the sequence of intensities at q = (0.28, 0.28) can be reproduced with a slightly smaller momentum (see dashed gray circle in Fig. 5b and dashed lines in Fig. 5c). This indicates that a small finetuning of the model parameters might give a full agreement between theory and experiment also at this momentum. Last but not least we note that the aforementioned sequence of intensities, and therefore the agreement with the theoretical results, is largely confirmed also when the experimental error bars are taken into account (see Supplementary Note 2).
Besides the overall agreement, there are two important discrepancies between the experiments and calculations. First, the experimentally observed sequence of intensities at q = (0.18, 0) is not reproduced by the calculations. Although the calculations do produce a crossing to a decreasing sequence upon doping at a relatively large momentum [q > (0.28, 0)], which satisfies the experimental results at q = (0.33, 0), they fail to reproduce the intensity sequence at the intermediate momentum q = (0.18, 0). The experimental results thus indicate this crossing should happen at a smaller momentum [q < (0.18, 0)]. Within the tJlike model and the parameter sets we considered in this work, the current choice with both \({t}^{{\prime} }\) and \(t^{\prime \prime}\) gives the best agreement (see discussion on Fig. 6 below). To fully reconcile this discrepancy, one may need to further finetune the Hamiltonian parameters, and in particular, consider their dopingdependence^{43}.
Second, the slope of the momentum dependence of the integrated experimental intensities is smaller compared with the calculated S(q). The slope extrapolates to a nonzero residual intensity at the Brillouin zone center q = (0, 0), making it gaplike, which is not found in the numerical results. The reason for this apparent disagreement is likely the interlayer interaction in this bilayer cuprate, which gives rise to the onset of the optical and acoustic magnon branches^{7,34,53,54}. While the 2D antiferromagnetic acoustic spin wave has a zero structure factor at the zone center [q = (0, 0)], the optical branch due to the interlayer coupling will show a nonzero intensity, thus leading to a ‘gap’ in the zone center. As our calculations do not include the coupling between the layers, this gap feature is absent. However, due to the interlayer interaction being much smaller than the intralayer one, the two branches quickly merge as q moves away from the zone center (≳0.1 r.l.u.)^{7,53,54}, and their total intensities will be dominated by the intralayer parameters. Therefore, the comparison between our calculations and the experiments is in reasonable agreement apart from the zone center.
The agreement between the t\({t}^{{\prime} }\)\(t^{\prime \prime}\)J model and experiments can be appreciated even more after looking at Fig. 6, where we present the comparison between the theoretical results of the three different tJlike models. Fig. 6a shows S(q) calculated for the ‘bare’ tJ model. Within this model, the sequence of intensities of the spin structure factor as a function of doping does not change in the experimental momentum range, unlike the results of the experiments. Fig. 6b shows the same quantity as Fig. 6a, but for the t\({t}^{{\prime} }\)J model. In this case, the calculated S(q) in the (π, π) direction shows the same qualitative behavior as in the experimental case. However, the spin structure factor S(q) in (π, 0) direction keeps increasing with doping in the studied momentum range, which is still inconsistent with the experiments. A different parameter choice with a larger value of \({t}^{{\prime} }\) in the t\({t}^{{\prime} }\)J model also does not improve the agreement between calculations and experiments (See Supplementary Fig. 4 and Note 4). The decreasing sequence of intensity upon doping in the (π, 0) direction is only achieved after including the thirdneighbor hopping \(t^{\prime \prime}\) as shown in Fig. 6c. However, it appears at relatively larger q than the experimental results. Nevertheless, it suggests the importance of longrange hoppings in achieving better agreement between experiment and theory. Further improvements may require finetuning of the parameters (see above).
Due to our real space approach, we have full control over the contribution of spin correlations of different range to the static spin structure factor. In fact, we can cut the summation in the Fourier transform [cf. Eqs. 5–6 in the Methods] to include only up to a certain number of neighbors. This analysis allows us to investigate the least effective range of magnetic correlations that can qualitatively produce the spin structure factor S(q) upon doping. In Fig. 7, we show the main results of this analysis on the t\({t}^{{\prime} }\)\(t^{\prime \prime}\)J model. Fig. 7a is a cartoon description of the real space spinspin correlations between one sample site around the centre of the 6 × 6 cluster and its neighbors up to thirdnearest ones. The color scale represents the value of such real space correlations. In Fig. 7b, we plot the spin structure factor S(q) calculated by including only up to thirdneighbor spinspin correlations. Since the shortrange correlations mostly account for the largeq dynamics, they become poor in sketching the smallq properties, which leaves an artificial gap around q = (0, 0). Except the smallq region, where the gap appears, the results show an ascending intensity as a function of doping in the intermediate q range in both the (π, π) and (π, 0) directions and a crossover to descending sequence in larger q, which qualitatively agree with the spin structure factor S(q) calculated from the full Fourier transform depicted in Fig. 7c. We also notice that almost all of the spectral weight of S(q) at (π, π) is already accounted for once solely the shortrange magnetic correlations up to the third neighbors are taken into account, cf. Figs. 5b and 7c. This is in stark contrast with the undoped case (see Supplementary Fig. 3): in the latter case, the spectral weight at (π, π) is strongly underestimated when only the shortrange magnetic correlations are taken into account. This is due to the importance of longrange correlations in the ordered antiferromagnet stabilized at halffilling. An indepth discussion of these properties, as well as a similar analysis with different range of correlations for the tJ and t\({t}^{{\prime} }\)J models can be found in the Supplementary Note 3.
Discussion
Our main experimental result is the unambiguous assessment of the momentum and doping evolution of the disentangled intrinsic spin and charge excitations measured by Cu L_{3}edge RIXS in doped Bi2212 samples. The disentangled spin responses show profiles that can be well fitted by a damped harmonic oscillator model, although they become over damped along (π, π) direction. The momentum and doping dependence of the spin and charge responses are different, implying the distinct nature of the two responses. In addition, the disentangled spin excitations well persist into the overdoped sample, which confirms that RIXS indeed observes the persistence of spin excitations in a large part of the Brillouin zone in doped cuprates.
The obtained experimental results on the spin excitations are qualitatively reproduced by numerical simulations of the tJlike model. It turns out that the bare tJ model is not enough and instead this model has to be supplemented by longerrange hoppings–which points out the decisive role of such hoppings in reproducing the experimentally observed paramagnons in doped cuprates. Furthermore, the extensive real space analysis shows that shortrange magnetic correlations are needed in order to cause the observed persistence of spin excitations in doped cuprates, meaning they need to be paramagnonic in nature. From that, two important consequences follow: On one hand, within the class of cuprate models with localized spins, those without any magnetic correlations at all seem not to be realistic for doped cuprates. (We note in passing that the class of models with localized spins is not only restricted to the studied t–J models. The Hubbard (or charge transfer) model description of the doped cuprates also possesses localized moments whose value is not substantially reduced compared with the t–J model case, since the number of doubly occupied sites is substantially suppressed in the Hubbardlike models with realistic values of the onsite Coulomb repulsion, cf. Fig. 1 of^{55} or Fig. 6 of^{56}.) On the other hand, this means that longerrange magnetic correlations do not play a crucial role in the doped cuprates. Altogether, this helps in resolving the paradox related to the persistence of the spin excitations upon doping the cuprates–despite a rapid collapse of the longrange magnetic correlations.
We also briefly comment on the charge excitations we obtained in the disentanglement analysis. With the same model calculations as for the spin structure factor, we can get the charge structure factor (see Supplementary Fig. 5), which can reproduce the increase in intensities as a function of doping in both the (π, 0) and (π, π) directions. However, the intensity anisotropy between the (π, 0) and (π, π) is missing. This might signal the onset of the bimagnons in the charge channel of the RIXS spectrum^{31,44} or suggest that the longerrange Coulomb interactions are important^{57}.
On the theory side, there are three important implications of the results presented in this paper: The first one is that this work shows that a recent theoretical suggestion that the spin excitations are responsible for the Tlinear dependence of the electronic scattering in the Hubbard model^{58} might indeed become a realistic scenario for the cuprates. This follows from the abovestated conclusion that, without any ambiguities, the collective spin excitations persist in the doped cuprates. The second one follows from the suggested crucial role played by the longerrange hoppings in the tJ models. Such a result goes in line with, inter alia, recent works advocating for the strong sensitivity of the phase diagram of the tJ like models to the value of the nextnearest neighbor hopping \({t}^{{\prime} }\)^{59,60} and thus ‘sweetens the bad news’ coming from the study suggesting the lack of superconductivity in the ground state of the 2D Hubbard model without longerrange hoppings^{61}. The third point relates to the fact that, as stated above, solely the shortrange magnetic correlations are needed to explain the persistence of the intensity of the paramagnons in doped cuprates, similar to the recently established welldefined magnons in the random tJ model up to 33% hole doping^{62}. Thus, an interesting task for theory would be to gain an intuitive understanding of why the shortrange magnetic correlation alone can lead to the lack of changes of the paramagnons along the (π, 0) direction.
We close the paper by presenting the impact of the results presented above on the issue of pairing mechanism in the cuprate superconductors. While there are many competing theories describing this issue, one of the widelyspread concepts suggests that the pairing is mediated by the magnetic excitations close to (π, π) momentum^{63}. More precisely, it was shown by inter alia Nocera et al.^{47} and Huang et al.^{64} that the spin fluctuations mediated pairing is mostly due to the lowenergy spin excitations with momentum q = (π, π) and that the ‘dopingpersistent’ highenergy magnetic excitations away from that momentum, i.e. the ones which are observed by RIXS, are not important to pairing. This finding has been corroborated by the experimental results of Meyers et al.^{22} and Dean et al.^{8}. Interestingly, here we have shown that even the q = (π, π) paramagnons are largely a result of the shortrange magnetic correlations. This is because the spectral weight in the magnetic response close to the q = (π, π) momentum is very similar both in the ‘full’ spin structure factor calculations as well as in the ones containing solely the shortrange magnetic correlations, see Fig. 6c versus Fig. 7. Note that this is in contrast to the undoped case, for which the peak around q = (π, π) is known to be hugely sensitive to the longerrange magnetic correlations (see Supplementary Fig. 3). Since the nature of the (π, π) paramagnons is central to any of the spin fluctuations mediated superconducting pairing mechanism, this finding plays an important role in our deeper understanding of the puzzle of superconductivity in cuprates. Finally, as a side note on the issue of pairing, the present study stresses the importance of the proper choice of the longerrange hoppings (\({t}^{{\prime} }\) and \(t^{\prime \prime}\)) in any realistic cuprate modeling. Thus, we believe that any model trying to explain superconductivity in the cuprates should include realistic values of these parameters as slight variations might qualitatively affect the observed physics.
Methods
RIXS Experiments and samples
The RIXS experiments were carried out with the SAXES spectrometer at the ADRESS beamline of the Swiss Light Source at the Paul Scherrer Institut^{65,66}. The incident Xray energy was set at the Cu L_{3} resonance peak at ~933 eV. The instrument resolution was determined by the elastic peak measured on carbon tape, giving an overall energy resolution of ~100 meV full width half maximum (FWHM). The singlecrystal samples of Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} were prepared by the floating zone method as described in ref. ^{67}. The samples were cleaved at base temperature to present fresh surfaces before the measurements. All the data were collected at base temperature ~24 K. The momentum transfer q is denoted in reciprocal lattice units (r. l. u.) using the pseudotetragonal unit cell with a = b = 3.82 Å.
Azimuthaldependent RIXS measurements
In twodimensional superconducting cuprates, the singlespinflip and nonspinflip excitations in the 3\({d}_{{x}^{2}{y}^{2}}\) orbital as well as the other dd excitations in the Cu L_{3}edge RIXS spectra show distinct geometry and polarization related characters, which are determined by their different scattering tensors^{35,68,69}. This allows to assess the nature of these excitations by either resolving the polarizations of both the incident and scattered photons^{70} or by azimuthaldependent RIXS measurements^{33}. In Fig. 1b, we show the scattering geometry and the sample rotation in the azimuthaldependent experiments. The directions of the incident and emitted xrays are fixed through the scattering angle to 130^{∘}, thus the total momentum transfer is fixed at q. Two linear polarizations (σ and π) are used for the incident xrays while the polarization of the emitted light is not resolved. The platelike sample is mounted on a wedged sample holder (with wedge angle θ_{w} = 10^{∘}, 20^{∘}, 40^{∘} and 50^{∘} in the experiments) to have a certain inplane momentum transfer. The azimuthal rotation axis is parallel to the total momentum transfer q, so that the projections of q in the sample reciprocal frame are unchanged during rotation, while the projections of the photon polarization are changing. This allows measuring the azimuthal dependence of the excitations at fixed momentum in the sample momentum space. When rotating the sample, the photon polarization will be continuously rotated in the sample space, and the scattering tensors will then result in different rotation dependences for different excitations.
Error estimations of the experimental data
The major errors of the RIXS spectra are the statistical errors of the photon counting, which are expressed as the square roots of total photon counts. Another error comes from the uncertainties in determining the zeroenergyloss positions in the spectra, which is done in examining the positions of the elastic peaks. Here we assume that this error is about ± 5% of the resolution, which is about ± 5 meV. This error is converted to the error of spectral intensity by multiplying the derivative of the spectrum. Other random errors are accounted for by the fitting errors (95% confidence interval) in the azimuthdependent fitting. All the above errors are treated as independent at each energyloss point and summed to a total error by the square root of their sum of squares. The errors of the integrated intensities are obtained by assuming that the errors of the decomposed spectral functions have an energy correlation defined by the resolution function. The error bars in the fitting of the damped harmonic oscillator (Fig. 4) are the fitting errors.
Fitting by damped harmonic oscillator model
The formula of the damped harmonic oscillator (DHO) model used for the fitting of the spin spectral funtions in Fig. 3 is:
A Gaussian resolution function with 100 meV FWHM is convoluted with the above DHO model to fit the results. The fitting energy range is [−0.8, 0.4] eV for all data except those with the smallest momenta (q = (0.09, 0) and (0.065, 0.065)), which are fitted in [−0.65, 0.4] eV. This is to reduce the influence of the long tail at highenergy loss.
DMRG calculations
In numerical calculations with OBC cluster, a proper correction is often added into the Hamiltonian to minimize the effects of missing terms at open edge. In this study, we introduced the edge factor to uniformize the mobility of charge.
The correct edge factor is calculated for each doping level and each different model (tJ, t\({t}^{{\prime} }\)J, t\({t}^{{\prime} }\)\(t^{\prime \prime}\)J). We extrapolate it by computing the dispersion δ = n_{in} − n_{out} where n_{in} is the averaged electron density taken over the sites which do not belong to the edges and n_{out} is the averaged electron density taken over the sites which form the edge of the cluster. We computed the dispersion δ for different values of the edge factor λ for each doping level and model and take the final value of the edge factor λ as that at which the dispersion δ = 0. Nevertheless, the obtained values (λ ~ 0.9 − 1.2) are close enough to 1 to smoothly connect the inside and edge of the cluster. Furthermore, by introducing this edge factor the Friedel oscillations as well as the most important finitesize effect coming from using OBC can be significantly reduced, as can be seen in the Supplementary Note 6.
We keep up to m = 7000 states in the DMRG calculations, leading to an error ϵ/N = 10^{−6}. We make sure that the local density in the system is isotropic by using the edge factor λ as described above and we compute the real space spinspin correlations 〈S_{i} ⋅ S_{j}〉 for all pairs (i, j) labeling the system’s sites. This way, we can compute the spin static factor S(q) as:
This method is only valid when the local zcomponent of the spins is small enough, e.g. \({S}_{i}^{z}\le 1{0}^{5}\) for all sites i. However, for certain doping levels and models, it was not possible to reach this level of accuracy for the local value of S^{z}. Therefore, we renormalized the S^{z}S^{z} correlations and computed S^{z}(q) as:
Due to the symmetries of the considered models, S(q) = 3S^{z}(q). Equation (6) was used for the doping level n = 0.22 for both the t\({t}^{{\prime} }\)J and the t\({t}^{{\prime} }\)\(t^{\prime \prime}\)J models (see Fig. 6).
Data availability
All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional raw experiment data to reproduce the analysis as well as data and scripts to reproduce the theoretical results are available in^{71}.
Code availability
The code for the numerical calculations will be made available from the corresponding authors upon reasonable request.
References
Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. & Zaanen, J. From quantum matter to hightemperature superconductivity in copper oxides. Nature 518, 179–186 (2015).
Lee, P. A., Nagaosa, N. & Wen, X.G. Doping a Mott insulator: Physics of hightemperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).
Xu, G. et al. Testing the itinerancy of spin dynamics in superconducting Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}. Nat. Phys. 5, 642–646 (2009).
Damascelli, A., Hussain, Z. & Shen, Z.X. Angleresolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys. 75, 473–541 (2003).
Tranquada, J. M. Spins, stripes, and superconductivity in holedoped cuprates. AIP Conf. Proc. 1550, 114–187 (2013).
Braicovich, L. et al. Magnetic excitations and phase separation in the underdoped La_{2−x}Sr_{x}CuO_{4} superconductor measured by resonant inelastic XRay scattering. Phys. Rev. Lett. 104, 077002 (2010).
Le Tacon, M. et al. Intense paramagnon excitations in a large family of hightemperature superconductors. Nat. Phys. 7, 725–730 (2011).
Dean, M. P. M. et al. Persistence of magnetic excitations in La_{2−x}Sr_{x}CuO_{4} from the undoped insulator to the heavily overdoped nonsuperconducting metal. Nat. Mater. 12, 1019–1023 (2013).
Dean, M. P. M. et al. Highenergy magnetic excitations in the cuprate superconductor Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}: Towards a unified description of its electronic and magnetic degrees of freedom. Phys. Rev. Lett. 110, 147001 (2013).
Dean, M. P. M. et al. Magnetic excitations in stripeordered La_{1.875}Ba_{0.125}CuO_{4} studied using resonant inelastic xray scattering. Phys. Rev. B 88, 020403(R) (2013).
Le Tacon, M. et al. Dispersive spin excitations in highly overdoped cuprates revealed by resonant inelastic XRay scattering. Phys. Rev. B 88, 020501(R) (2013).
Ishii, K. et al. Highenergy spin and charge excitations in electrondoped copper oxide superconductors. Nat. Commun. 5, 3714 (2014).
Lee, W. S. et al. Asymmetry of collective excitations in electron and holedoped cuprate superconductors. Nat. Phys. 10, 883–889 (2014).
Dean, M. P. M. et al. Itinerant effects and enhanced magnetic interactions in Bibased multilayer cuprates. Phys. Rev. B 90, 220506(R) (2014).
Guarise, M. et al. Anisotropic softening of magnetic excitations along the nodal direction in superconducting cuprates. Nat. Commun. 5, 5760 (2014).
Minola, M. et al. Collective nature of spin excitations in superconducting cuprates probed by resonant inelastic XRay scattering. Phys. Rev. Lett. 114, 217003 (2015).
Wakimoto, S. et al. Highenergy magnetic excitations in overdoped La_{2−x}Sr_{x}CuO_{4} studied by neutron and resonant inelastic XRay scattering. Phys. Rev. B 91, 184513 (2015).
Peng, Y. Y. et al. Magnetic excitations and phonons simultaneously studied by resonant inelastic XRay scattering in optimally doped Bi_{1.5}Pb_{0.55}Sr_{1.6}La_{0.4}CuO_{6+δ}. Phys. Rev. B 92, 064517 (2015).
Ellis, D. S. et al. Correlation of the superconducting critical temperature with spin and orbital excitations in (Ca_{x}La_{1−x})(Ba_{1.75−x}La_{0.25+x})Cu_{3}O_{y} as measured by resonant inelastic XRay scattering. Phys. Rev. B 92, 104507 (2015).
Huang, H. Y. et al. Raman and fluorescence characteristics of resonant inelastic XRay scattering from doped superconducting cuprates. Sci Rep 6, 19657 (2016).
Monney, C. et al. Resonant inelastic XRay scattering study of the spin and charge excitations in the overdoped superconductor La_{1.77}Sr_{0.23}CuO_{4}. Phys. Rev. B 93, 075103 (2016).
Meyers, D. et al. Doping dependence of the magnetic excitations in La_{2−x}Sr_{x}CuO_{4}. Phys. Rev. B 95, 075139 (2017).
Ivashko, O. et al. Damped spin excitations in a doped cuprate superconductor with orbital hybridization. Phys. Rev. B 95, 214508 (2017).
Minola, M. et al. Crossover from collective to incoherent spin excitations in superconducting cuprates probed by detuned resonant inelastic XRay scattering. Phys. Rev. Lett. 119, 097001 (2017).
Chaix, L. et al. Resonant inelastic XRay scattering studies of magnons and bimagnons in the lightly doped cuprate La_{2−x}Sr_{x}CuO_{4}. Phys. Rev. B 97, 155144 (2018).
Peng, Y. Y. et al. Dispersion, damping, and intensity of spin excitations in the monolayer (Bi,Pb)_{2}(Sr,La)_{2}CuO_{6+δ} cuprate superconductor family. Phys. Rev. B 98, 144507 (2018).
Robarts, H. C. et al. Anisotropic damping and wave vector dependent susceptibility of the spin fluctuations in La_{2−x}Sr_{x}CuO_{4} studied by resonant inelastic XRay scattering. Phys. Rev. B 100, 214510 (2019).
James, A. J. A., Konik, R. M. & Rice, T. M. Magnetic response in the underdoped cuprates. Phys. Rev. B 86, 100508(R) (2012).
Benjamin, D., Klich, I. & Demler, E. Singleband model of resonant inelastic XRay scattering by quasiparticles in highT_{c} cuprate superconductors. Phys. Rev. Lett. 112, 247002 (2014).
KanászNagy, M., Shi, Y., Klich, I. & Demler, E. A. Resonant inelastic XRay scattering as a probe of band structure effects in cuprates. Phys. Rev. B 94, 165127 (2016).
Jia, C., Wohlfeld, K., Wang, Y., Moritz, B. & Devereaux, T. P. Using RIXS to uncover elementary charge and spin excitations. Phys. Rev. X 6, 021020 (2016).
Tsutsui, K. & Tohyama, T. Incidentenergydependent spectral weight of resonant inelastic XRay scattering in doped cuprates. Phys. Rev. B 94, 085144 (2016).
Kang, M. et al. Resolving the nature of electronic excitations in resonant inelastic XRay scattering. Phys. Rev. B 99, 045105 (2019).
Peng, Y. Y. et al. Influence of apical oxygen on the extent of inplane exchange interaction in cuprate superconductors. Nat. Phys. 13, 1201–1206 (2017).
Ament, L. J. P., van Veenendaal, M., Devereaux, T. P., Hill, J. P. & van den Brink, J. Resonant inelastic XRay scattering studies of elementary excitations. Rev. Mod. Phys. 83, 705–767 (2011).
Bisogni, V. et al. Bimagnon studies in cuprates with resonant inelastic XRay scattering at the O K edge. I. Assessment on La_{2}CuO_{4} and comparison with the excitation at Cu L_{3} and Cu K edges. Phys. Rev. B 85, 214527 (2012).
Devereaux, T. P. & Hackl, R. Inelastic light scattering from correlated electrons. Rev. Mod. Phys. 79, 175–233 (2007).
Sugai, S., Suzuki, H., Takayanagi, Y., Hosokawa, T. & Hayamizu, N. Carrierdensitydependent momentum shift of the coherent peak and the LO phonon mode in ptype highT_{c} superconductors. Phys. Rev. B 68, 184504 (2003).
Sugai, S. et al. Superconducting pairing and the pseudogap in the nematic dynamical stripe phase of La_{2−x}Sr_{x}CuO_{4}. J. Phys.: Condens. Matter 25, 475701 (2013).
Chao, K. A., Spałek, J. & Oleś, A. M. Canonical perturbation expansion of the Hubbard model. Phys. Rev. B 18, 3453–3464 (1978).
Delannoy, J.Y. P., Gingras, M. J. P., Holdsworth, P. C. W. & Tremblay, A.M. S. Lowenergy theory of the \({t}{t^{\prime}}{t^{\prime\prime}}{U}\) hubbard model at halffilling: Interaction strengths in cuprate superconductors and an effective spinonly description of La_{2}CuO_{4}. Phys. Rev. B 79, 235130 (2009).
Sénéchal, D. & Tremblay, A.M. S. Hot spots and pseudogaps for hole and electrondoped hightemperature superconductors. Phys. Rev. Lett. 92, 126401 (2004).
Drozdov, I. K. et al. Phase diagram of Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} revisited. Nat. Commun. 9, 5210 (2018).
Jia, C. J. et al. Persistent spin excitations in doped antiferromagnets revealed by resonant inelastic light scattering. Nat. Commun. 5, 3314 (2014).
Eder, R., Ohta, Y. & Maekawa, S. Anomalous spin and charge dynamics of the tJ model at low doping. Phys. Rev. Lett. 74, 5124–5127 (1995).
Tohyama, T., Horsch, P. & Maekawa, S. Spin and charge dynamics of the tJ model. Phys. Rev. Lett. 74, 980–983 (1995).
Nocera, A., Patel, N. D., Dagotto, E. & Alvarez, G. Signatures of pairing in the magnetic excitation spectrum of strongly correlated twoleg ladders. Phys. Rev. B 96, 205120 (2017).
Tohyama, T., Mori, M. & Sota, S. Dynamical density matrix renormalization group study of spin and charge excitations in the \(t{{\mbox{}}}\,{{\mbox{}}}{t}^{{}^{{\prime} }}{{\mbox{}}}\,{{\mbox{}}}J\) fourleg ladder. Phys. Rev. B 97, 235137 (2018).
Pärschke, E. M. et al. Numerical investigation of spin excitations in a doped spin chain. Phys. Rev. B 99, 205102 (2019).
Jiang, S., Scalapino, D. J. & White, S. R. Groundstate phase diagram of the \(t{t}^{{\prime} }J\) model. Proc. Natl. Acad. Sci. USA 118, e2109978118 (2021).
Gong, S., Zhu, W. & Sheng, D. N. Robust dwave superconductivity in the squarelattice t − J model. Phys. Rev. Lett. 127, 097003 (2021).
Jiang, H.C., Weng, Z.Y. & Kivelson, S. A. Superconductivity in the doped t − J model: Results for fourleg cylinders. Phys. Rev. B 98, 140505 (2018).
Reznik, D. et al. Direct observation of optical magnons in YBa_{2}Cu_{3}O_{6.2}. Phys. Rev. B 53, R14741(R) (1996).
Hayden, S. M., Aeppli, G., Perring, T. G., Mook, H. A. & Doğan, F. Highfrequency spin waves in YBa_{2}Cu_{3}O_{6.15}. Phys. Rev. B 54, R6905(R) (1996).
Moreo, A., Scalapino, D. J., Sugar, R. L., White, S. R. & Bickers, N. E. Numerical study of the twodimensional Hubbard model for various band fillings. Phys. Rev. B 41, 2313–2320 (1990).
Kung, Y. F. et al. Doping evolution of spin and charge excitations in the Hubbard model. Phys. Rev. B 92, 195108 (2015).
Fidrysiak, M. & Spałek, J. Unified theory of spin and charge excitations in highT_{c} cuprate superconductors: A quantitative comparison with experiment and interpretation. Phys. Rev. B 104, L020510 (2021).
Wú, W., Wang, X. & Tremblay, A.M. NonFermi liquid phase and linearintemperature scattering rate in overdoped twodimensional Hubbard model. Proc. Natl. Acad. Sci. U.S.A. 119, e2115819119 (2022).
Jiang, H.C. & Devereaux, T. P. Superconductivity in the doped Hubbard model and its interplay with nextnearest hopping \({t}^{{\prime} }\). Science 365, 1424–1428 (2019).
Jiang, Y.F., Zaanen, J., Devereaux, T. P. & Jiang, H.C. Ground state phase diagram of the doped hubbard model on the fourleg cylinder. Phys. Rev. Research 2, 033073 (2020).
Qin, M. et al. Absence of superconductivity in the pure twodimensional Hubbard model. Phys. Rev. X 10, 031016 (2020).
Kumar, A., Sachdev, S. & Tripathi, V. Quasiparticle metamorphosis in the random t − J model. Phys. Rev. B 106, L081120 (2022).
Scalapino, D. J. A common thread: the pairing interaction for unconventional superconductors. Rev. Mod. Phys. 84, 1383–1417 (2012).
Huang, E. W., Scalapino, D. J., Maier, T. A., Moritz, B. & Devereaux, T. P. Decrease of dwave pairing strength in spite of the persistence of magnetic excitations in the overdoped Hubbard model. Phys. Rev. B 96, 020503 (2017).
Ghiringhelli, G. et al. SAXES, a high resolution spectrometer for resonant XRay emission in the 4001600ev energy range. Rev. Sci. Instrum. 77, 113108 (2006).
Strocov, V. N. et al. Highresolution soft Xray beamline ADRESS at the Swiss Light Source for resonant inelastic Xray scattering and angleresolved photoelectron spectroscopies. J. Synchrotron Rad. 17, 631–643 (2010).
Weyeneth, S., Schneider, T. & Giannini, E. Evidence for KosterlitzThouless and threedimensional XY critical behavior in Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}. Phys. Rev. B 79, 214504 (2009).
Ament, L. J. P., Ghiringhelli, G., Sala, M. M., Braicovich, L. & van den Brink, J. Theoretical demonstration of how the dispersion of magnetic excitations in cuprate compounds can be determined using resonant inelastic XRay scattering. Phys. Rev. Lett. 103, 117003 (2009).
Haverkort, M. W. Theory of resonant inelastic XRay scattering by collective magnetic excitations. Phys. Rev. Lett. 105, 167404 (2010).
Fumagalli, R. et al. Polarizationresolved Cu L_{3}edge resonant inelastic XRay scattering of orbital and spin excitations in NdBa_{2}Cu_{3}O_{7−δ}. Phys. Rev. B 99, 134517 (2019).
Zhang, W. et al. Unravelling the nature of the spin excitations disentangled from the charge contributions in a doped cuprate superconductor. Zenodo https://zenodo.org/record/7286412#.Y5nuVnaZNPY (2022).
Acknowledgements
The experiments were performed at the ADRESS beamline of the Swiss Light Source at the Paul Scherrer Institut (PSI). The experimental work at PSI is supported by the Swiss National Science Foundation through project no. 200021_178867, and the Sinergia network Mott Physics Beyond the Heisenberg Model (MPBH) (SNSF Research Grants CRSII2_160765/1 and CRSII2_141962). The theoretical work is supported by the Narodowe Centrum Nauki (NCN, Poland) under Project Nos. 2016/22/E/ST3/00560 (C.E.A. and K.W.), 2021/40/C/ST3/00177 (C.E.A.) and 2016/23/B/ST3/00839 (K.W.). We acknowledge support by the Interdisciplinary Centre for Mathematical and Computational Modeling (ICM), University of Warsaw (UW), within grant no G814. S.N. acknowledges support from SFB 1143 project A05 (projectid 247310070) of the Deutsche Forschungsgemeinschaft. T.C.A. acknowledges funding from the European Union’s Horizon 2020 research and innovation program under the Marie SkłodowskaCurie grant agreement No. 701647 (PSIFELLOWII3i program). We acknowledge the valuable discussions with Jonathan Pelliciari, Mingu Kang, Riccardo Comin and Ekaterina Pärschke. We kindly thank U. Nitzsche for technical support. For the purpose of Open Access, the author has applied a CCBY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission
Author information
Authors and Affiliations
Contributions
W.Z. and C.E.A. contributed equally to this work. W.Z. and T.S. conceived the project. K.W. conceived the theory approach. W.Z., Y.T., T.C.A., E.P. and T.S. performed the RIXS experiments with the support of V.N.S. C.E.A. performed the DMRG calculations with the support of S.N. E.G. grew and characterized the single crystals. W.Z. and C.E.A. analyzed the data in discussion with T.S., S.N., and K.W. W.Z., C.E.A., K.W., and T.S. wrote the manuscript with the input from all authors. T.S. and K.W. coordinated the research.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
41535_2022_528_MOESM1_ESM.pdf
Supplementary Materials for Unraveling the Nature of Spin Excitations Disentangled From Charge Contributions in a Doped Cuprate Superconductor
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Zhang, W., Agrapidis, C.E., Tseng, Y. et al. Unraveling the nature of spin excitations disentangled from charge contributions in a doped cuprate superconductor. npj Quantum Mater. 7, 123 (2022). https://doi.org/10.1038/s41535022005285
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41535022005285
This article is cited by

Unraveling the nature of spin excitations disentangled from charge contributions in a doped cuprate superconductor
npj Quantum Materials (2022)