Damped spin excitations in a doped cuprate superconductor with orbital hybridization

TL;DR

This study uses RIXS to investigate overdamped spin excitations in underdoped LSCO, revealing the significant impact of orbital hybridization on zone-boundary dispersion and contrasting with undoped cuprates.

Contribution

It demonstrates that orbital hybridization between $d_{x^2-y^2}$ and $d_{z^2}$ states significantly influences spin excitation dispersion in doped cuprates, extending beyond single-band model predictions.

Findings

Overdamped spin excitations show strong dispersion along certain directions.

Orbital hybridization enhances zone-boundary dispersion.

Hybridization effects differ from undoped cuprates.

Abstract

A resonant inelastic x-ray scattering (RIXS) study of overdamped spin-excitations in slightly underdoped La$_{2-x}$Sr$_{x}$CuO$_4$ (LSCO) with $x=0.12$ and $0.145$ is presented. Three high-symmetry directions have been investigated: (1) the antinodal $(0,0)\rightarrow (1/2,0)$, (2) the nodal $(0,0)\rightarrow (1/4,1/4)$ and (3) the zone boundary direction $(1/2,0)\rightarrow (1/4,1/4)$ connecting these two. The overdamped excitations exhibit strong dispersions along (1) and (3), whereas a much more modest dispersion is found along (2). This is in strong contrast to the undoped compound La$_{2}$CuO$_4$ (LCO) for which the strongest dispersions are found along (1) and (2). The $t-t^{\prime}-t^{\prime\prime}-U$ Hubbard model used to explain the excitation spectrum of LCO predicts $-$ for constant $U/t$ $-$ that the dispersion along (3) scales with $(t^{\prime}/t)^2$. However, the diagonal hopping $t^{\prime}$ extracted on LSCO using single-band models is low ($t^{\prime}/t\sim-0.16$) and decreasing with doping. We therefore invoked a two-orbital ($d_{x^2-y^2}$ and $d_{z^2}$) model which implies that $t^{\prime}$ is enhanced. This effect acts to enhance the zone-boundary dispersion within the Hubbard model. We thus conclude that hybridization of $d_{x^2-y^2}$ and $d_{z^2}$ states has a significant impact on the zone-boundary dispersion in LSCO.

Tables

Table 1. Table 1: Parametrization – using the Hubbard model – of the spin-excitation dispersion of LCO  Coldea  et al. ( 2001 ); Headings  et al. ( 2010 ) , LSCO x = 0.12 𝑥 0.12 x=0.12 (this work) and Bi 2201 2201 2201   Peng  et al. ( 2016 ) . (★) Values obtained from the fit using the same procedure as described in Appendix B, which thus can be directly compared.

La2-xSrxCuO4	$U$ (eV)	$U/t$	$t^{\prime }/t$	$t^{\prime \prime }/t$	$Z$	Ref.
$x=0$	2.2	7.4	0	0	$1.18$	Coldea et al.,2001; Headings et al.,2010
$x=0$	3.6	8.3	-0.313	0.167	1.219	Delannoy et al.,2009
$x=0$	3.9	9.1	-0.308	0.154	1.219	★
$x=0.12$	2.9	6.8	-0.405	0.202	1.219	★
Bi$2201$
$x=0$	3.4	8.0	-0.352	0.176	1.219	★

Equations

$${\chi }^{\prime \prime }(\omega )$$

$$=\frac{{\chi }_0^{\prime \prime }}{2{\omega }_1}\left[\frac{\gamma /2}{(\omega -{\omega }_1{)}^2+(\gamma /2{)}^2}-\frac{\gamma /2}{(\omega +{\omega }_1{)}^2+(\gamma /2{)}^2}\right],$$

$$\frac{E_{ZB}}{12Z{J}_2}\approx 1+\frac{1}{12}\left[112-{\left(\frac{U}{t}\right)}^2\right]{\left(\frac{t^{\prime }}{t}\right)}^2.$$

$$\omega (\overline{q})=Z\sqrt{A(\overline{q}{)}^2-B(\overline{q}{)}^2}$$

$$A(\overline{q})=A_0+A_1+\dots \&B(\overline{q})=B_0+B_1+\dots$$

$$P_j(h,k)=\cos jha+\cos jka$$

$$X_j(h,k)=\cos jha\cdot \cos jka$$

$$X_{3a}(h,k)=\cos 3ha\cdot \cos ka+\cos ha\cdot \cos 3ka,$$

$$A_0=2J_1\&B_0=-J_1{P}_1$$

$$A_1=J_2\left(-26-8X_1+P_2\right)\&B_1=16J_2{P}_1$$

$$A(\overline{q})=A_0+A_1+A_0^{\prime }+A_0^{\prime \prime }+A_c^{\prime }+A_1^{\prime }+A_c^{\prime \prime }+A_1^{\prime \prime }$$

$$B(\overline{q})=B_0+B_1+B_c^{\prime }.$$

$$A_0^{\prime }=2J_1^{\prime }(X_1-1)\&A_0^{\prime \prime }=J_1^{\prime \prime }(P_2-2)$$

$$A_c^{\prime }=-\frac{8J_1}{U^2}\left(-t^{\prime 2}+4t^{\prime }{t}^{\prime \prime }-2t^{\prime \prime 2}\right)(P_2-2)$$

$$B_c^{\prime }=-\frac{4J_1}{U^2}\left[\left(6t^{\prime 2}-4t^{\prime }{t}^{\prime \prime }\right)(X_1-1)+3t^{\prime \prime 2}(P_2-2)\right]P_1$$

$$A_1^{\prime }=2J_2^{\prime }(X_2+4X_1-2P_2-1)$$

$$A_c^{\prime \prime }=\frac{2J_1^{\prime }{J}_1^{\prime \prime }}{U}\left(-3X_2+2X_1+5P_2-X_{3a}-7\right)$$

$$A_1^{\prime \prime }=J_2^{\prime \prime }(P_4-8X_2+4P_2-2)$$

$$\frac{E_{ZB}}{12Z{J}_2}\approx 1+\frac{1}{12}\left(112-\frac{J_1}{J_2}\right){\left(\frac{t^{\prime }}{t}\right)}^2.$$

$$\begin{gathered} \frac{U}{t}\ge \sqrt{\frac{28+112{\left(\frac{t^{\prime }}{t}\right)}^2}{2+3{\left(\frac{t^{\prime }}{t}\right)}^2}}, \\ \text{and}\left|\frac{t^{\prime }}{t}\right|\lesssim \mathrm{0.686.} \end{gathered}$$

Full text

I Introduction

Considerable research is being undertaken in the quest to reach consensus on the mechanism of high-temperature superconductivity Lee et al. (2006) and the associated pseudogap phase Norman et al. (2005) in copper-oxide materials (cuprates). The energy scales governing the physical properties of these layered materials therefore remain of great interest. It is known that these materials are characterized by a strong superexchange interaction $J_{1}=4t^{2}/U$ where $t$ is the nearest-neighbor hopping integral and $U$ is the Coulomb interaction. To first order, this energy scale sets the bandwidth of the spin-excitation spectrum. Resonant inelastic x-ray scattering (RIXS) experiments Ament et al. (2011) have demonstrated that this bandwidth stays roughly unchanged across the entire phase diagram Tacon et al. (2011); M. P. M. Dean et al. (2013) of hole doped cuprates. It has also been demonstrated that the cuprates belong to a regime (of $t$ and $U$) where the second-order exchange interaction $J_{2}=4t^{4}/U^{3}$ contributes to a spin-excitation dispersion along the antiferromagnetic zone boundary (AFZB) Coldea et al. (2001); Headings et al. (2010); Delannoy et al. (2009); Dalla Piazza et al. (2012). Moreover, it is known from band structure calculations and experiments that the next nearest-neighbor (diagonal) hopping integral $t^{\prime}$ constitutes a non-negligible fraction of $t$ Yoshida et al. (2006). Empirically Pavarini et al. (2001), the superconducting transition scales with the ratio $t^{\prime}/t$ whereas Hubbard type models predict the opposite trend White and Scalapino (1999); Maier et al. (2000). As a resolution, a two-orbital model – in which hybridization of $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ states suppresses $T_{c}$ and enhances $t^{\prime}$ – has been put forward Sakakibara et al. (2010).

Here, we address the question as to how $t^{\prime}$ influences the spin-excitation spectrum at, and in the vicinity of, the antiferromagnetic zone boundary. We have therefore studied – using the RIXS technique – slightly underdoped compounds of La2-xSrxCuO4 (LSCO) with $x=0.12$ and $0.145$. Even though the system is not antiferromagnetically ordered at these dopings, we quantify the zone-boundary dispersion $\omega(\overline{q})$ by $E_{ZB}=\omega(\nicefrac{{1}}{{2}},0)-\omega(\nicefrac{{1}}{{4}},\nicefrac{{1}}{{4}})$. In doped LSCO a strongly enhanced zone-boundary dispersion is observed. As will also be shown, within the $t-t^{\prime}-t^{\prime\prime}-U$ Hubbard model, one generally expects that the zone-boundary dispersion scales with $t^{\prime}/t$ with a prefactor that depends on $U/t$. The Fermi-surface topology of LSCO, obtained from photoemission spectroscopy and analyzed with a single-band tight binding model, suggests that $t^{\prime}$ decreases with increasing doping Yoshida et al. (2006); Chang et al. (2008a). The Hubbard model is thus within a single-band picture not consistent with the experiment. However, using a two-orbital model, hybridization between $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ states enhances $t^{\prime}$ Sakakibara et al. (2010). This provides a satisfactory description of the zone-boundary dispersion. We thus conclude that the two-orbital model Sakakibara et al. (2010) is necessary to understand the spin-excitation spectrum of doped LSCO.

II Method

The RIXS experiment was carried out at the ADvanced RESonant Spectroscopies (ADRESS) beamline Ghiringhelli et al. (2006); Strocov et al. (2010) at the Swiss Light Source (SLS) with the geometry shown in Fig. 1(h). The newly installed CARVING RIXS manipulator allowed us to probe the full kinematically accessible reciprocal space $\overline{q}=(h,k)$ with a scattering angle of $130^{\circ}$. Incident photons with an energy of $933$ eV (at the Cu $L_{3}$-edge resonance) gave an instrumental energy and momentum resolution of $132$ meV and $0.01$ Å*-1* respectively. Both the linear horizontal (LH) and linear vertical (LV) light polarizations were applied to probe high quality single crystals of La2-xSrxCuO4 with $x=0.12$ and $0.145$ ($T_{c}=27$ and $35$ K respectively). These crystals were grown by the traveling floating zone method Komiya et al. (2002) and previously characterized in neutron Chang et al. (2009); Rømer et al. (2013); Chang et al. (2012) and muon spin-resonance ($\mu$SR) Chang et al. (2008b) experiments. Ex situ prealignment of the samples was carried out using a Laue diffractometer. The samples were cleaved in situ using a standard top-post technique and all data were recorded at $T=20$ K. Although being in the low temperature orthorhombic (LTO) crystal structure, tetragonal notation $a{\cong}b\approx 3.78$ Å($c\approx 13.2$ Å) is adopted to describe the in-plane momentum $(h,k)$ in reciprocal lattice units $2\pi/a$.

III Results

Fig. 1(a-c) displays grazing exit RIXS spectra of La1.88Sr0.12CuO4 recorded with incident LH light polarization along three trajectories as indicated in (g). Data along the same directions but measured with incident LV polarization are shown in (d-f). Besides the strong elastic scattering found at the specular condition $[Q=(0,0)]$, an elastic charge-density-wave (CDW) reflection is found – consistent with existing literature Thampy et al. (2014); Croft et al. (2014) – along the $(h,0)$ direction at $Q_{\mathrm{CDW}}=(\delta_{1},\delta_{2})$ with $\delta_{1}=0.24(6)$ and $\delta_{2}\simeq 0.01$. The charge order reflection serves as a reference point, demonstrating precise alignment of the crystal.

For grazing exit geometry, it has previously been demonstrated that spin excitations are enhanced in the LH channel Tacon et al. (2011). In Fig. 2(a,b), selected raw RIXS spectra recorded with LH polarization are shown for momenta near the $(\nicefrac{{1}}{{2}},0)$ and $(\nicefrac{{1}}{{4}},\nicefrac{{1}}{{4}})$ points. The low-energy part of the spectrum consists of three components: a weak elastic contribution, a smoothly varying background and a damped spin excitation. It is immediately clear that the excitations near $(\nicefrac{{1}}{{4}},\nicefrac{{1}}{{4}})$ are significantly softened compared to those observed around the $(\nicefrac{{1}}{{2}},0)$-point (see Fig. 2(a,b)).

For a more quantitative analysis of the magnon dispersion, we modeled the elastic line with a Gaussian for which the standard deviation $\sigma=56$ meV was set by the instrumental energy resolution. A second order polynomial function is used to mimic the background. Finally, to analyze the spin excitations we adopted the response function of a damped harmonic oscillator Tacon et al. (2011); Monney et al. (2016); Lamsal and Montfrooij (2016):

[TABLE]

where the damping coefficient $\gamma/2=\sqrt{\omega_{0}^{2}-\omega_{1}^{2}}$. The RIXS intensities are modeled by $\left[n_{B}(\omega)+1\right]\chi^{\prime\prime}(\omega)$, where $n_{B}(\omega)=\left[\exp(\hbar\omega/k_{B}T)-1\right]^{-1}$ is the Bose factor. As shown in Fig. 2(a-b), fitting to this simple model provides a good description of the observed spectra. In this fashion, we extracted the spin-excitation pole dispersion $\omega_{1}(\overline{q})$ (Fig. 2(c-e)) along the three trajectories shown in the inset. To avoid the influence of CDW ordering on the spin-excitation dispersion Miao et al. (2017), we analyzed around the charge ordering vector spectra of LSCO $x=0.145$ where charge order is absent.

The extracted spin-excitation dispersion of LSCO $x=0.12$ and $0.145$ is to be compared with the magnon dispersion of the parent compound La2CuO4 Braicovich et al. (2009, 2010); Coldea et al. (2001); Headings et al. (2010). Along the antinodal $(\nicefrac{{1}}{{2}},0)$ direction comparable dispersions are found. This is consistent with the weak doping dependence reported on LSCO M. P. M. Dean et al. (2013) and the YBa2Cu3O7-δ (YBCO) system Tacon et al. (2011). For the nodal $(\nicefrac{{1}}{{4}},\nicefrac{{1}}{{4}})$ direction, the dispersion of the doped compound is, however, strongly softened compared to La2CuO4. Whereas this effect has been reported for Bi-based Guarise et al. (2014); Dean et al. (2014) and overdoped LSCO Monney et al. (2016), we demonstrate directly by an azimuthal scan how exactly this softening appears. Notice that the azimuthal dependence is closely related (but not exactly identical) to the scan along the antiferromagnetic zone boundary.

IV Discussion

A recent systematic study Peng et al. (2016) of undoped cuprate compounds concluded that the zone-boundary dispersion scales with the crystal field splitting $E_{{z^{2}}}$ of the $d_{x^{2}-y^{2}}$ and $d_{z^{2}}$ states. Exact numerical determination of $E_{{z^{2}}}$ is still a matter of debate Sakakibara et al. (2010); Hozoi Liviu et al. (2011). For a tetragonal system, $E_{{z^{2}}}$ generally depends on the ratio between copper to apical and planar oxygen distances Sala et al. (2011). The crystal field splitting $E_{z^{2}}$ can in principle be accessed by measuring the $dd$ excitations. For LCO, interpretations of the $dd$ excitations have consistently placed the $d_{z^{2}}$ level above (i.e. closer to the Fermi level) both the $d_{xz,yz}$ and $d_{xy}$ states Sala et al. (2011); Peng et al. (2016). This is also consistent with density functional theory (DFT) Sakakibara et al. (2010) and ab initio Hozoi Liviu et al. (2011) calculations of the electronic band structure that find the $d_{z^{2}}$-band above the $t_{2g}$ states. In doped LSCO $x=0.12$, the spectral weight of the $dd$ excitations is redistributed and the ”center of mass” is shifted to lower energies (see Fig. 3). The $d_{xy}$ states are expected to be the least sensitive to crystal field changes Sala et al. (2011). Therefore, it is conceivable that the $d_{xz,yz}$ and $d_{z^{2}}$ states are shifting to lower energies. Again from DFT calculations (see Appendix C), we expect the $d_{z^{2}}$ states to appear above those of $d_{xz,yz}$. Our experimental results thus (Fig. 3) suggest that the crystal field splitting $E_{{z^{2}}}$ in doped LSCO $x=0.12$ is smaller compared to LCO. Yet, the zone-boundary dispersion is larger in LSCO $x=0.12$ (Fig. 2). The present experiment is therefore not lending support for a correlation between the zone-boundary dispersion and the crystal field splitting $E_{{z^{2}}}$.

The spin-excitation dispersion of doped LSCO is analyzed using an effective Heisenberg Hamiltonian derived from a $t-t^{\prime}-t^{\prime\prime}-U$ Hubbard model Delannoy et al. (2009); Headings et al. (2010); Dalla Piazza et al. (2012). This discussion has three steps. First, an approximative analytical expression for the zone-boundary dispersion is derived. Next, we compare to the experimentally obtained results using the known single-band tight-binding values of $t,t^{\prime}$ and $t^{\prime\prime}$. It is shown that this approach leads to unrealistically low values of the Coulomb interaction $U$. The $d_{z^{2}}$ band is therefore included. This two-orbital scenario allows us to describe the zone boundary dispersion with more realistic input parameters, as presented in the last part of the discussion.

The simplest version of the Hubbard model contains only three parameters: the Coulomb interaction $U$, the band width ($4t$), and a renormalization factor $Z$ – known to have little momentum dependence. To lowest order in $J_{1}=4t^{2}/U$, no magnon dispersion is expected along the zone boundary. Therefore, to explain the zone-boundary dispersion – first observed on La2CuO4 – higher order terms $J_{2}=4t^{4}/U^{3}$ were included Coldea et al. (2001); Headings et al. (2010) to the model. Later, it has been pointed out that higher-order hopping terms $t^{\prime}$ and $t^{\prime\prime}$ can also contribute significantly Delannoy et al. (2009); Dalla Piazza et al. (2012). Generally, the effective Heisenberg model yields a dispersion Delannoy et al. (2009); Dalla Piazza et al. (2012) $\omega(\overline{q})=Z\sqrt{A(\overline{q})^{2}-B(\overline{q})^{2}}$ where $A(\overline{q})$ and $B(\overline{q})$ – given in the Appendix A – are depending on $U,t,t^{\prime}$ and $t^{\prime\prime}$. The zone boundary dispersion can be quantified by $E_{ZB}=\omega(\nicefrac{{1}}{{2}},0)-\omega(\nicefrac{{1}}{{4}},\nicefrac{{1}}{{4}})$. Using the single-band Hubbard model with realistic parameters Yoshida et al. (2006); Pavarini et al. (2001); Delannoy et al. (2009) ($U/t\sim 8$, $\lvert t^{\prime}\rvert\leq t/2$ and $t^{\prime\prime}=-t^{\prime}/2$) for hole doped cuprates, we find (see Appendix A):

[TABLE]

A key prediction is thus that $E_{ZB}$ scales as $(t^{\prime}/t)^{2}$ with a pre-factor that depends on $(U/t)^{2}$.

This effective Heisenberg model is in principle not applicable to doped and hence antiferromagnetically disordered cuprates. For an exact description of the data, more sophisticated numerical methods have been developed C. J. Jia et al. (2014). However, in the absence of analytical models, the Heisenberg model serves as a useful effective parametrization tool to describe the damped spin excitations. Within a single-band tight-binding model, angle resolved photoemission spectroscopy (ARPES) experiments have found that $t^{\prime}$ decreases slightly with increasing doping Yoshida et al. (2006); Chang et al. (2008a). The stronger zone-boundary dispersion can thus not be attributed to an increase of $t^{\prime}$. Parameterizing the doping dependent zone-boundary dispersion would thus imply a strong renormalization of $U$ with increasing doping. For example, if we set $4t=1720$ meV (obtained from local density approximation (LDA) and ARPES Pavarini et al. (2001); Fatuzzo et al. (2014); J. Chang et al. (2013)) and $t^{\prime}/t=-0.16$ and $t^{\prime\prime}=-t^{\prime}/2$, a fit yields $U/t\sim 5$ and $Z\sim 0.7$. Although these parameters provide a satisfactory description of the dispersion, the values of $U$ and $Z$ are not physically meaningful.

This failure combined with the observation of a reduced level splitting between the $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ states (Fig. 3) motivates a two-band model. It has been demonstrated that $d_{z^{2}}$ states contribute to effectively increase the $t^{\prime}$ hopping parameter Sakakibara et al. (2010). Keeping $Z=1.219$ as in La2CuO4 Delannoy et al. (2009) and $t^{\prime\prime}=-t^{\prime}/2$, a satisfactory description (solid line in Fig. 2) of the spin-excitation dispersion is obtained for $t^{\prime}/t=-0.405$ and $U/t=6.8$. Notice that a similar ratio of $t^{\prime}/t$ has previously been inferred from the rounded Fermi-surface topology of Tl2Ba2CuO6+x Platé et al. (2005); Peets et al. (2007); Palczewski et al. (2008) a material for which the $d_{z^{2}}$ states are expected to be much less important Sakakibara et al. (2012). It could thus suggest that $t^{\prime}/t\approx-0.4$ is common to single layer cuprates but masked in LSCO due to the repulsion between the $d_{x^{2}-y^{2}}$ and $d_{z^{2}}$ bands that pushes the van Hove singularity close to the Fermi level and effectively reshapes the Fermi-surface topology Sakakibara et al. (2010). The more realistic values of $U$ and $Z$, suggest that – for LSCO – the two-orbital character of this system is an important ingredient to accurately describe the spin-excitation spectrum.

Once having extracted $U/t$ and $t^{\prime}/t$ by fitting the experimental spin-excitation spectrum, we plot – in Fig. 4 – the normalized zone-boundary dispersion $E_{ZB}/(12ZJ_{2})$ versus $\frac{1}{12}\left(t^{\prime}/t\right)^{2}\left[112-\left(U/t\right)^{2}\right]$. The same parameters were extracted (see Table 1 in the Appendix) from published RIXS data on La2CuO4 and Bi2Sr0.99La1.1CuO6+δ Peng et al. (2016) and plotted in Fig. 4. All three compounds follow approximately the predicted correlation between $E_{ZB}/(12ZJ_{2})$ and $\frac{1}{12}\left(t^{\prime}/t\right)^{2}\left[112-\left(U/t\right)^{2}\right]$. This suggests that the zone-boundary dispersion is controlled by the parameters $t^{\prime}/t$ and $U/t$. It would be interesting to extend this parametrization to include higher doping concentrations of LSCO. However, from existing RIXS data on overdoped single crystals of LSCO it is not possible to perform the analysis presented here Wakimoto et al. (2015); Monney et al. (2016). For LSCO $x=0.23$, for example, the zone-boundary dispersion has not been measured Monney et al. (2016).

Finally, we notice that recent RIXS experiments on LSCO thin films using SrLaAlO4 (SLAO) substrates found a much less pronounced softening of the spin-excitation dispersion around the $(\nicefrac{{1}}{{4}},\nicefrac{{1}}{{4}})$ point Meyers et al. (2017). A possible explanation is that LSCO films on SLAO have a larger $c$-axis lattice parameter and hence also a larger copper to apical-oxygen distance than what is found in bulk crystals Locquet et al. (1998); Abrecht et al. (2003). As a consequence, the $d_{z^{2}}$ states are less relevant and hence lead to a less pronounced zone-boundary dispersion.

V Conclusions

In summary, a comprehensive RIXS study of underdoped LSCO $x=0.12$ and $0.145$ were presented. The spin-excitation dispersion was studied along three high-symmetry directions and a strong zone-boundary dispersion is reported. The spin-excitation dispersion was parametrized and discussed using a Heisenberg Hamiltonian derived from a Hubbard model including higher-order hopping integrals. Within this model, the zone-boundary dispersion scales with next-nearest-neighbor hopping integral $t^{\prime 2}$. We argue that hybridization between $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$, which is especially strong in LSCO, leads to an enhanced $t^{\prime}$. This effect – consistent with the observations – leads to a stronger zone-boundary dispersion within the $t-t^{\prime}-t^{\prime\prime}-U$ Hubbard model.

VI Acknowledgments:

We acknowledge support by the Swiss National Science Foundation under grant No. BSSGI0$\_155873$ and through the SINERGIA network Mott Physics Beyond the Heisenberg Model. This work was performed at the ADRESS beamline of the SLS at the Paul Scherrer Institut, Villigen PSI, Switzerland. We thank the ADRESS beamline staff for technical support. M.D. and T.S. have been partially funded by the Swiss National Science Foundation within the D-A-CH program (SNSF Research Grant No. $200021$L $141325$). X.L. acknowledges financial support from the European Community’s Seventh Framework Program (FP$7/2007-2013$) under grant agreement No. $290605$ (COFUND: PSI-FELLOW). D.M.N. was supported by the Swiss National Science Foundation under Grant No. $51$NF$40$_$141828$ through the NCCR-MARVEL. C.M. acknowledges support from the Swiss National Science Foundation under grant $PZ00P2\_154867$.

VII Appendix A

Here we describe the spin-excitation dispersion of the Heisenberg Hamiltonian derived from the $t-t^{\prime}-t^{\prime\prime}-U$ Hubbard model in two steps. We first consider the simplest model where $t^{\prime}=t^{\prime\prime}=0$ before including higher-order hopping terms.

Generally the dispersion takes the form:

[TABLE]

where $Z$ is a renormalization factor and $\overline{q}=(h,k)$. When the Hubbard model contains only the nearest-neighbor hopping integral $t$, we expand $A(\overline{q})$ and $B(\overline{q})$ to second order in $t$:

[TABLE]

To express $A_{i}$ and $B_{i}$, we define: $J_{1}=\frac{4t^{2}}{U}$ and $J_{2}=\frac{4t^{4}}{U^{3}}$. Moreover we set:

[TABLE]

[TABLE]

[TABLE]

where $j=1,2,3,$ or $4$. With this notation we have:

[TABLE]

and

[TABLE]

When the zone-boundary dispersion is defined by $E_{ZB}=\omega(\nicefrac{{1}}{{2}},0)-\omega(\nicefrac{{1}}{{4}},\nicefrac{{1}}{{4}})$, one finds $E_{ZB}=12ZJ_{2}$. Therefore, a zone-boundary dispersion is only found when second-order terms $J_{2}$ are included. Notice also that since $P_{1}(\nicefrac{{1}}{{2}},0)=P_{1}(\nicefrac{{1}}{{4}},\nicefrac{{1}}{{4}})=0$, the $B$-terms are not contributing to the zone-boundary dispersion.

Now let us include second-nearest $t^{\prime}$ and third-nearest-neighbor $t^{\prime\prime}$ hopping integrals. This involves several additional contributions to $A(\overline{q})$ and $B(\overline{q})$:

[TABLE]

[TABLE]

To express these new terms, we introduce the following notation $J^{\prime}_{1}=\frac{4t^{\prime 2}}{U}$, $J^{\prime}_{2}=\frac{4t^{\prime 4}}{U^{3}}$, $J^{\prime\prime}_{1}=\frac{4t^{\prime\prime 2}}{U}$ and $J^{\prime\prime}_{2}=\frac{4t^{\prime\prime 4}}{U^{3}}$. Geometrically the following contributions correspond to different hopping path combinations including the cyclic ones.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

As $B^{\prime}_{c}$ scales with $P_{1}$, it is again found that $B(\overline{q})$ does not contribute to the zone-boundary dispersion. In Fig. 5, we show the numerical evaluation of $E_{ZB}$ for realistic values of $U/t$, $t^{\prime}/t$ and with $t^{\prime\prime}=-t^{\prime}/2$. When neglecting terms scaling with $J^{\prime}_{2}$, $J^{\prime\prime}_{2}$ and $J^{\prime}_{1}J^{\prime\prime}_{1}$, only Eq. 10 and 11 contribute. Using $P_{2}(\nicefrac{{1}}{{2}},0)=2$, $P_{2}(\nicefrac{{1}}{{4}},\nicefrac{{1}}{{4}})=-2$, $X_{1}(\nicefrac{{1}}{{2}},0)=-1$ and $X_{1}(\nicefrac{{1}}{{4}},\nicefrac{{1}}{{4}})=0$, we find:

[TABLE]

This approximation is valid as long as:

[TABLE]

As shown in Fig. 5, this analytical expression is a good approximation to the full numerical calculation. Thus it is justified to neglect terms scaling with $J^{\prime}_{2}$, $J^{\prime\prime}_{2}$ and $J^{\prime}_{1}J^{\prime\prime}_{1}$ for a realistic cuprate values of $U/t$ and $t^{\prime}/t$.

VIII Appendix B

Now, having derived the spin-excitation dispersion within the $t-t^{\prime}-t^{\prime\prime}-U$ Hubbard model, it is possible to fit the experimentally observed dispersion. A final comment goes to the prefactor $Z$. It is found that, including higher-order hopping integrals $t^{\prime}$ and $t^{\prime\prime}$, $Z$ has a slowly varying momentum dependence. To simplify our analysis we used the mean value obtained Delannoy et al. (2009) in the first Brillouin zone for the half filled compound La2CuO4. We thus have $Z=1.219$ constant. From ARPES Fatuzzo et al. (2014); J. Chang et al. (2013) experiments and LDA calculations Pavarini et al. (2001) we have that $t=0.43$ eV and $t^{\prime\prime}=-t^{\prime}/2$. Our fitting parameters are thus $U$ and $t^{\prime}$. In this fashion we obtain a good description of the spin-excitation dispersion of LCO and LSCO $x=0.12$ (see Fig. 2 in the main text). The obtained values are given in Table 1. In Fig. 6 and Table 1, we display in addition our fit and associated fit parameters from the spin-excitation spectrum measured on Bi$2201$ (Ref. Peng et al., 2016). With these values of $U$ and $t^{\prime}$, the relation – shown in Fig. 4 – between $E_{ZB}$ and $t^{\prime}$ is established.

IX Appendix C

To guide our intuition of how the $d_{z^{2}}$ states evolve as a function of doping, we have carried out DTF calculations of the LSCO band structure as a function of doping. These calculations were performed using the WIEN2K package Blaha et al. (2001) in the LTO crystal structure. The doping dependence of the electronic structure for LSCO was approximated by a rigid band shift of all Cu $d$ orbitals in order to obtain the correct $d$-shell filling. For every calculated doping value, the experimentally derived crystal structure has been used Radaelli et al. (1994). In the calculation, the Kohn-Sham equation is solved self-consistently by using a full-potential linear augmented plane wave (LAPW) method on a uniform grid of $12\times 12\times 12$ $k$ points in the Brillouin zone. The exchange-correlation term is treated within the generalized gradient approximation (GGA) in the parametrization of Perdew, Burke and Enzerhof (PBE) Perdew et al. (1996). The plane-wave cutoff condition was set to $RK_{max}=7$ where $R$ is the radius of the smallest LAPW sphere ($i.e.$ $1.63$ bohrs) and $K_{max}$ denotes the plane-wave cutoff. Fig. 7 shows the orbital and atomic resolved band structure and density of states (DOS) of LSCO in the tetragonal crystal structure. As shown in panel (a), the $d_{z^{2}}$ derived band disperses in a binding energy range of $E-E_{F}=-1.3~{}eV$ close to $\Gamma$ and $E-E_{F}=-0.3~{}eV$ at $M$. The orbital resolved DOS of the $d_{z^{2}}$ band has a peak at $E-E_{F}=-0.5~{}eV$, while closer to $E_{F}$ the $d_{z^{2}}$ DOS rapidly decays. This peak originates from the flat shape of the $d_{z^{2}}$ band close to $M$. Therefore to track the doping dependence of the $d_{z^{2}}$ energy level, the position of the band at the $M$ point is plotted as a function of doping $x$ in Fig. 7(c). With increasing doping $x$ the $d_{z^{2}}$ energy level approaches the Fermi energy. Note that our DFT calculation agrees with recently published results obtained by ab initio calculations Sakakibara et al. (2010).