Abstract
How coherent quasiparticles emerge by doping quantum antiferromagnets is a key question in correlated electron systems, whose resolution is needed to elucidate the phase diagram of copper oxides. Recent resonant inelastic Xray scattering (RIXS) experiments in holedoped cuprates have purported to measure highenergy collective spin excitations that persist well into the overdoped regime and bear a striking resemblance to those found in the parent compound, challenging the perception that spin excitations should weaken with doping and have a diminishing effect on superconductivity. Here we show that RIXS at the Cu L_{3}edge indeed provides access to the spin dynamical structure factor once one considers the full influence of light polarization. Further we demonstrate that highenergy spin excitations do not correlate with the doping dependence of T_{c}, while lowenergy excitations depend sensitively on doping and show ferromagnetic correlations. This suggests that highenergy spin excitations are marginal to pairing in cuprate superconductors.
Introduction
Initial Cu L_{3}edge resonant inelastic Xray scattering (RIXS) measurements on undoped and weakly underdoped cuprates^{1,2} complemented earlier neutron and Raman scattering experiments, seeming to favour a spinfluctuation scenario as a viable explanation of superconductivity^{3}. However, more recent RIXS measurements on overdoped cuprates^{3,4,5,6} have shown persistent highenergy spin excitations to very high doping levels where superconductivity disappears. In contrast neutron and Raman measurements display an absence of robust spin excitations in the overdoped regime^{7,8}; this conflict undermines an understanding of unconventional superconductivity in the cuprates, making an investigation of how spin excitations manifest in the RIXS crosssection a crucial component to its resolution.
In this letter, we reconcile these seemingly incompatible experimental results by computing Cu L_{3}edge RIXS spectra using exact diagonalization (ED), capable of reproducing major experimental features. We demonstrate with light polarization analysis that the RIXS crosssection in a crossedpolarization geometry can be interpreted simply in terms of the spin dynamical structure factor S(q, ω), which enables a comparison between different scattering experiments. Utilizing determinant quantum Monte Carlo (DQMC), we study in detail the momentum and doping dependence of S(q, ω), finding strong changes near both (π, π) and (0, 0) with relatively insensitive antiferromagnetic zone boundary (AFZB) paramagnons upon hole doping. Moreover, with electron doping these same AFZB paramagnons harden significantly, which has recently been confirmed experimentally. Underlying this observed behaviour is a framework of local spin exchange, which remains robust even with significant doping away from the parent antiferromagnet. In contrast, our calculations show a sensitive evolution of lowenergy paramagnons near (0, 0) and (π, π), which give evidence for the predominance of ferromagnetic correlations. These results highlight the importance of spectral weight and dispersion at low energies in establishing a relevant energy scale and strength of spin fluctuations for pairing rather than higherenergy AFZB paramagnons.
Results
The relationship between RIXS and S(q, ω)
RIXS is a resonant technique and its sensitivity to magnetic excitations arises as a result of corelevel spinorbit interactions in the intermediate state (see Fig. 1a). Although in Mott insulators it has been shown that the RIXS crosssection can be approximated by S(q, ω) when the charge excitations are gapped, it is not clear whether the same approximation carries over to doped systems where the ground state is no longer that of a Mott insulator with commensurate filling^{9,10}.
To answer this question, we numerically evaluate the RIXS crosssection as a function of momentum (Fig. 1c) directly from the KramersHeisenberg formula^{11,12} using small cluster ED of an effective singleband Hubbard Hamiltonian (including both nearest t and nextnearest neighbour hopping t′ and onsite Coulomb repulsion U) at various electron concentrations n; details are given in the Methods section. Figure 1c displays the RIXS spectra calculated for the experimental geometry discussed in ref. 3. The RIXS spectra, even without outgoing polarization discrimination, agree well with S(q, ω) for different electron concentrations at the chosen momentum space points accessible on the same finite size clusters for each calculation. This is particularly true at halffilling where the charge gap ensures that only spin excitations can be visible in the given energy range. The main differences occur in the doped systems at higher energy (close to 2t), which are of less interest for our spin analysis. The important result shown in Fig. 1c concerns the suppression of these higher energy peaks in the crosspolarized geometry (see Fig. 1c π−σ RIXS) which leads to a significant improvement in the comparison between the RIXS crosssection and S(q, ω). This indicates that Cu L_{3}edge RIXS with crossed polarizations (a fourparticle correlator) provides access to the spin excitation spectrum encoded in S(q, ω) (a twoparticle correlator) for doped as well as undoped cuprates. (To further confirm this agreement between RIXS and S(q, ω) for more momentum points, we manually adjusted the Cu Ledge energy to be 1.8*930 eV=1674, eV so that momentum points up to (π, π) can be reached. For more details see Supplementary Note 1.)
The momentum and doping dependence of S(q, ω)
Having established a relationship between RIXS and S(q, ω), we focus now on the momentum and doping dependence of S(q, ω) for the singleband Hubbard model. Here we employ the numerically exact DQMC method (see Methods) with maximum entropy analytic continuation on larger lattices with fine control of the electron concentration through the chemical potential. As shown in Fig. 2 the DQMC calculations qualitatively reproduce both the momentum and doping evolution of the RIXS measurements found in refs 1, 2, 3, 4, 5, 6. A comparison between the intensity and dispersion of lowenergy magnetic excitations near (0, 0) and (π, π) shows a transition from antiferromagnetic to ferromagnetic spin correlations with increasing doping. [In an antiferromagnetic system, the dynamical spin structure factors show gapless excitations at both (0, 0) and (π, π), with strong intensity around (π, π). In a ferromagnetic system, the dynamical spin structure factors show strong intensity with gapless excitations at (0, 0) and much weaker intensity with gapped excitations when approaching (π, π)]. However, the spectra of higherenergy AFZB paramagnons show relatively little change with hole doping other than a general decrease of intensity, suggesting that spin excitations do not soften even in the heavily overdoped regime. For electron doping, AFZB paramagnons surprisingly harden by 50% at 15% doping that has been observed recently in the prototypical electrondoped cuprate Nd_{2−x}Ce_{x}CuO_{4} (ref. 13). For additional analysis and discussion, see Supplementary Discussion.
Theory to understand AFZB paramagnons
The behaviour of these AFZB paramagnons stands in stark contrast to naive expectations of spin softening with either hole or electron doping: (i) longrange AF order collapses quickly upon doping with a small (intermediate) concentration of holes (electrons), and one would expect spin excitations to soften accordingly^{14,15}; (ii) shortrange AF correlations are further weakened due to a dilution of AF bonds (see Fig. 3a) in the locally static spin picture. Indeed, the nearest neighbour spinspin correlations from the DQMC calculations with electron doping decrease in a manner surprisingly well described by a locally static spin picture where the doped electrons are immobile, as shown in Fig. 3b.
We can address these points by considering the role of threesite exchange^{16}, which lowers the system energy when both doped carriers and AF correlations are present (see Fig. 3a). A local spin flip in an otherwise AF background produces a ferromagnetic alignment of nearest neighbour spins, which costs additional energy (of the same order as the spin exchange J=4t^{2}/U) by suppressing hole (or doubleoccupancy) delocalization represented by the threesite terms. In fact, if we consider only the spin exchange contributions, the combined energy of a singlespin flip in the doped system (breaking both spin exchange and threesite bonds) is larger than that of the undoped system by ~J/4 (see Fig. 3a). This hardening has been observed in ED calculations of S(q, ω) for the H_{Hubbard} and H_{t−J}+H_{3s} Hamiltonians (see Methods) as shown in Fig. 3c upon electron doping.
The situation is more subtle with hole doping, because this ‘locally static model’ no longer completely applies as seen in Fig. 3b. The negative nextnearest neighbour hopping t′ (positive for electron doping) promotes magnetic sublattice mixing and a much larger destruction of the AF correlations^{17}. With hole doping the trend observed in RIXS is fully recovered only in the Hubbard model, as shown in Fig. 2, implying that higher order processes absent in t−Jtype models become crucial in quantitatively reproducing the spin wave dispersion^{18}.
Discussion
How do these results reconcile the seemingly contradictory observations between RIXS, neutron and Raman scattering? First, inelastic neutron scattering probes spin excitations particularly well around (π, π) momentum transfer, showing a vanishing spectral weight in the regime of large hole doping p≃0.3 (refs 7, 19). This behaviour is also visible in the numerical results presented in Fig. 2a, which suggest that the impact of doping on the intensity and dispersion of excitations near (0, 0) and (π, π) is not symmetric. The decreasing correlation length with doping, evidenced by the spin gap at (π, π) and the weak dispersion towards (π/2, π/2), thus impacts these momentum points more strongly than the AFZB paramagnons, in accordance with a locally static picture. Second, Raman scattering^{8,20,21} shows a softening of the socalled bimagnon (double spinflip or twomagnon) response upon both hole and electron doping. This trend has been reproduced by our ED calculations of the B_{1g} Raman response shown in Fig. 4 (see Methods for the calculation details and the verification of bimagnon peaks). Strong magnonmagnon interactions reduce the bimagnon Raman peak energy from twice that of the singlemagnon bandwidth as determined by AFZB magnons and quickly reduce the overall intensity. Taken as a whole, our results provide a qualitative, and in some cases quantitative, agreement with the salient experimental features of neutron scattering, Raman and RIXS measurements, suggesting that coherent propagating spin waves quickly disappear with the destruction of longrange AF order upon doping, while shortrange, single spinflip processes can survive to high doping levels as reflected in the evolution of S(q, ω).
Full polarization control will allow RIXS to become an effective tool for directly observing spin dynamics along the AFZB, particularly noting the electron/hole doping differences. Together with the domeshaped superconducting phase diagram, these results imply that AFZB spin fluctuations might play a relatively minor role in the pairing mechanism, consistent with established experimental and numerical observations^{22,23,24}. This calls into question a simple view of pairing that emphasizes only the spin exchange energy scale J. However, we suggest that a definitive resolution to this issue would come from future RIXS experiments along the BZ diagonal (out to (π/2, π/2)) to illuminate the evolution from antiferro to ferromagnetic correlations, compare with neutron scattering results and ultimately shed additional light on the intriguing mystery of cuprate hightemperature superconductivity.
Methods
Numerical techniques
We use exact diagonalization (ED) to evaluate the RIXS crosssection from the Kramers–Heisenberg formula^{11}, spin dynamical structure factor S(q, ω) and Raman scattering crosssection^{25} on small clusters with periodic boundary conditions. We employ a 12site Betts cluster^{26} in evaluating the RIXS crosssection and S(q, ω) shown in Fig. 1. The Raman scattering response shown in Fig. 4 has been evaluated on 16 and 18site square (or diamondshaped) clusters, and the 18site cluster was employed to evaluate S(q, ω) for H_{Hubbard}, H_{t−J} and H_{t−J}+H_{3s} shown in Fig. 3. The ED calculations for H_{Hubbard} are performed with the Parallel ARnoldi PACKage (PARPACK) and the crosssections obtained by use of the biconjugate gradient stabilized method and continued fraction expansion^{12}. The ED calculations on H_{t−J} and H_{t−J}+H_{3s} models are performed using the Lanczos algorithm. Finite temperature DQMC simulations^{27,28} were performed on H_{Hubbard} to obtain the imaginary time spinspin correlation function from which the real frequency response function S(q, ω) was obtained by analytic continuation using the maximum entropy method (MEM)^{29,30}. These simulations were performed on 8 × 8 square lattice clusters with periodic boundary conditions at an inverse temperature β=3/t for the same Hubbard Hamiltonian parameter values utilized in the ED studies. For this set of parameters, the DQMC method exhibits a significant fermion sign problem^{31} over the entire doping range, which we address in the MEM^{30} (see Supplementary Methods and Supplementary Fig. 6). MEM requires the use of a model function for determining an entropic prior in the analytic continuation routine. We utilize a Lorentzian model whose peak as a function of q is determined from a simple spin wave dispersion at small q out to the AFZB; however, beyond the AFZB the model assumes no softening as expected for longrange antiferromagnetism with the top of the magnon band set by approximations for the spin exchange J and an assumed reduction of the spin moment by quantum fluctuations. While some quantitative changes occur with significant changes to these default models, we have checked that the qualitative behaviour remains robust. The MEM routine returns the real frequency spin susceptibility from which S(q, ω) is obtained from the fluctuationdissipation theorem. More details about the models and numerical algorithms can be found in the following Methods and Supplementary Methods.
RIXS
The Cu L_{3}edge RIXS crosssection is calculated using the Kramers–Heisenberg formula^{11} for the singleband Hubbard model
in which
where q is the momentum transfer; ω_{in} and ω=ω_{in}−ω_{out} are the incident photon energy (in our study the Cu L_{3}edge) and photon energy transfer, respectively; E_{0} is the ground state energy of the system in the absence of a corehole; 0› is the ground state wave function; (and h.c.) dictates the dipole transition process from Cu 2p to the 3d level (or from Cu 3d to 2p), with the Xray polarization either π or σ (the polarization vector parallel or perpendicular to the scattering plane); and Γ is the inverse corehole lifetime (see Supplementary Note 2). In H_{Hubbard}, <…> and … represent a sum over the nearest and nextnearest neighbour sites, respectively. The Hamiltonian for the intermediate state also involves the onsite energy for creating a 2p core hole, Coulomb interaction U_{c} induced by the corehole and spinorbit coupling λ, all denoted as in H_{CH}. represents the spinorbital coupling coefficients. The angle between the incident and the scattered photon propagation vectors is set to be 50°. The parameters used in the RIXS calculation are t=0.4 eV, U=8t=3.2 eV, t′=−0.3t=−0.12 eV, , U_{c}=−4t=−1.6 eV, λ=13 eV and Γ=1t=0.4 eV (refs 32, 33). RIXS spectra at halffilling are taken only at the Cu L_{3} resonance, and upon doping at the resonance closest to the halffilling Cu L_{3}edge resonant energy. The RIXS results were obtained for a Lorentzian broadening with half width at half maximum (HWHM)=0.01 eV (0.025t) and a Gaussian broadening with HWHM=0.047 eV (0.118t) on the energy transfer. The spin dynamical structure factor S(q, ω), discussed in the next section, for H_{Hubbard} was calculated using the same parameters to make comparison to our RIXS results.
Spin dynamical structure factor
The spin dynamical structure factor is defined as
where we have studied the H_{Hubbard}, H_{t−J} and H_{t−J}+H_{3s} Hamiltonians:
E_{0} is the corresponding ground state energy of the model Hamiltonian; for H_{Hubbard}; , for H_{t−J} and H_{3s}; ; and is restricted in the subspace without double occupancy and , in which the operator annihilates a dressed electron whose hopping conserves the number of effective doubly occupied sites^{34}. To explore the similarities and differences between H_{Hubbard} and H_{t−J} (with and without H_{3s}), we calculate S(q, ω) on the three model Hamiltonians with the parameters J=0.4t, t′=−0.25t and U=10t (corresponding to J=0.4t by the relation J=4t^{2}/U).
Locally static model
In the ‘locally static model’ (see Fig. 3b) it is assumed that the holes destroy the shortrange spinspin correlations solely by the effect of ‘static’ doping, that is, by removing the spins and thus cutting spin bonds. In this case, the nearest neighbour spinspin correlation can be calculated in the following way:
where 〈S_{0}S_{1}› is the abbreviation of 〈0S_{i}S_{j}0› for two neighbouring sites i and j, and p is the concentration of either doped holes (P=1−n) or doped electrons (P=n−1).
Raman scattering
We calculate the Raman scattering crosssection in the B_{1g} channel using the nonresonant response function for H_{Hubbard}^{25}:
in which ?(k)=−2t(cosk_{x}+cosk_{y})−4t′cosk_{x}cosk_{y} is the bare band dispersion, with parameters U=8t and t′=−0.3t. t is taken as 0.4 eV to make comparison with experimental data.
This twoparticle response also has been studied recently in cluster dynamical meanfield theory^{35} showing that, if calculated fully gauge invariantly, the nonresonant Raman B_{1g} response shows the presence of a strong bimagnon peak at half filling. Nevertheless, the Raman spectrum calculated using this method for doped systems is sensitive to both charge and spin excitations in the lowenergy regime. Our identification of the bimagnon excitations in the Raman spectra relies primarily on the qualitative evolution of the peaks in agreement with experimental observations^{20}. We note that all of the excitations visible in our Raman spectra correspond to ΔS=0 transitions that have B_{1g} symmetry. At halffilling, the energy of the excitation to which we assign bimagnon character lies within the charge gap, which makes the bimagnon assignment clear. Upon either hole or electron doping, we expect to develop charge excitations in the Raman response at low energy and for the twomagnon response to soften and decrease in intensity. Our assignment corresponds to an upper bound for the bimagnon energy scale with doping where the additional structure at low energies signals either charge excitations or a substantial broadening of the bimagnon excitations now represented by multiple features in the ED result (as discussed in connection with comparisons between DQMC and ED results). However, the energy scale clearly softens and, more importantly, the intensity drops (significantly) in agreement with the experimental observations where the bimagnon ‘peak’ becomes nearly impossible to distinguish from the charge background almost immediately upon crossing the AFM phase boundary.
Additional information
How to cite this article: Jia, C. J. et al. Persistent spin excitations in doped antiferromagnets revealed by resonant inelastic light scattering. Nat. Commun. 5:3314 doi: 10.1038/ncomms4314 (2014).
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Acknowledgements
We thank G. Ghiringhelli, R. Hackl, J. P. Hill, B. J. Kim, M. Le Tacon, W.S. Lee and J. Tranquada for discussions. This work was supported at SLAC and Stanford University by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering, under Contract No. DEAC0276SF00515 and by the Computational Materials and Chemical Sciences Network (CMCSN) under Contract No. DESC0007091. C.J.J. is also supported by the Stanford Graduate Fellows in Science and Engineering. C.C.C. is supported by the Aneesur Rahman Postdoctoral Fellowship at Argonne National Laboratory, operated under the U.S. Department of Energy Contract No. DEAC0206CH11357. T.T. is supported by the GrantinAid for Scientific Research (Grant No. 22340097) and Strategic Programs for Innovative Research (SPIRE), the Computational Materials Science Initiative (CMSI) from MEXT. Y.F.K. was supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1147470. T.T. and T.P.D. acknowledge the YIPQS program of YITP, Kyoto University. A portion of the computational work was performed using the resources of the National Energy Research Scientific Computing Center (NERSC) supported by the U.S. Department of Energy, Office of Science, under Contract No. DEAC0205CH11231.
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C.J.J. developed the computer codes and performed the RIXS, S(q, ω), and Raman scattering exact diagonalization calculations using the Hubbard Hamiltonian and C.C. C. assisted in the development of exact diagonalization computer codes. T.T. performed the S(q, ω) calculations using H_{t−J} with and without H_{3s}. E.A.N. and B.M. performed the determinant quantum Monte Carlo and analytic continuation calculations on the Hubbard Hamiltonian to evaluate S(q, ω) and developed computer codes together with S.J. Y.F.K. performed analysis on determinant quantum Monte Carlo data. K.W. developed the ‘locally static model’ approach. K.W., C.J.J., B.M., C.C.C., E.A.N. and T.P.D. wrote the manuscript. T.P.D. and B.M. are responsible for project planning.
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Jia, C., Nowadnick, E., Wohlfeld, K. et al. Persistent spin excitations in doped antiferromagnets revealed by resonant inelastic light scattering. Nat Commun 5, 3314 (2014). https://doi.org/10.1038/ncomms4314
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