Reciprocity between local moments and collective magnetic excitations in the phase diagram of BaFe$_2$(As$_{1-x}$P$_x$)$_2$

Unconventional superconductivity arises at the border between the strong coupling regime with local magnetic moments and the weak coupling regime with itinerant electrons, and stems from the physics of criticality that dissects the two. Unveiling the nature of the quasiparticles close to quantum criticality is fundamental to understand the phase diagram of quantum materials. Here, using resonant inelastic x-ray scattering (RIXS) and Fe-K$_\beta$ emission spectroscopy (XES), we visualize the coexistence and evolution of local magnetic moments and collective spin excitations across the superconducting dome in isovalently-doped BaFe$_2$(As$_{1-x}$P$_x$)$_2$ (0.00$\leq$x$\leq0.$52). Collective magnetic excitations resolved by RIXS are gradually hardened, whereas XES reveals a strong suppression of the local magnetic moment upon doping. This relationship is captured by an intermediate coupling theory, explicitly accounting for the partially localized and itinerant nature of the electrons in Fe pnictides. Finally, our work identifies a local-itinerant spin fluctuations channel through which the local moments transfer spin excitations to the particle-hole (paramagnons) continuum across the superconducting dome.


INTRODUCTION
It is now well established that unconventional superconductivity (SC) originates from a unique and augmented manifestation of the electronic correlation properties that arises as the system is driven towards a quantum critical region via various tuning parameters such as doping and pressure [1][2][3][4][5][6][7][8][9][10][11] . On one side of the superconducting dome, the correlation strength is strongly enhanced, leading to diverse quantum many-body effects such as the Mott insulating state, non-Fermi-liquid behaviour, and magnetic orders (see Fig. 1a,b) 3,4,[12][13][14] . On the other side of the dome, the correlation strength is often substantially suppressed, and the low-energy physics can be described by a more conventional Fermi-liquid theory 11 . The transition region between these two limits of correlation holds a quantum critical point barely understood. Generally, SC is optimized in this intermediate region where the cooperation of a strongly enhanced non-Fermi-liquid behaviour and the presence of a quantum criticality leads to unconventional and not understood physics 11 . This generic phase diagram suggests that intertwined electronic and magnetic instabilities, arising from the intermediate correlation strength close to this critical region, induce strong Cooper pairing 3,4,15 .
The physics which determines the evolution of the spin excitations is crucially dependent on the interaction strength of the electrons compared to their bandwidth. We explain this phenomenon schematically in Fig. 1c-g. The non-interacting density of states (DOS) of electrons ( Fig. 1c) is renormalized by the interaction in the two extreme limits as: (i) the bands become sharper in the weak coupling quasiparticle picture (Fig. 1d) and (ii) split into two Mott-like bands characterized by local moments in the strong coupling limit (Fig. 1e).
(i) In the weak coupling limit, the low-energy spin excitations (χ ii , see Fig. 1d) are very fragile and mix with the particle-hole continuum, failing to form a localized moment. (ii) In the strong coupling limit, the spin excitations across the Mott-bands (χ ll , see Fig. 1e) feature a gapped behaviour on the order of the onsite energy U without the particle-hole continuum 13,14 . However, as the correlation strength is tuned to the intermediate coupling region (Fig. 1f ), the correlated DOS co-hosts the quasiparticle DOS at low-energy and Mott states at high-energy. In this way a new spin-excitation channel (χ il see Fig. 1g) appears, through which the local moment can now decay to the particle-hole (paramagnon) channels across the magnetic quantum critical point (QCP). Fe pnictides (FePns) can be placed in this intermediate region where the interplay of local and itinerant electronic states leads to high temperature SC 1,2,11,15 . An important open question in this context is how local magnetic moments and collective spin excitations are evolving across the superconducting dome in FePns.
FePns have a layered structure that reduces dimensionality and the parent compounds exhibit a spin-density wave with collinear antiferromagnetic (AF) order (see Fig. 1b), which gives way to SC via doping as outlined in Fig. 1a [1][2][3][4][5][6][7][8][9][10][11] . The origin of the magnetism is poised between being itinerant, as in Cr, and localized, as in cuprates or heavy fermion materials 1,2,5-11,16 producing an uncommon behaviour which has important consequences for the properties of FePns. Theoretical models proposed that the pairing interaction leading to the superconducting phase is provided by residual AF fluctuations persisting upon doping 3,4,15 . Yet, owing to the contribution of the AF fluctuations from local and itinerant states, a complex interplay between them arises, which is believed to play a crucial role in shaping the superconducting dome. Thus, the experimental study of magnetism across the phase diagram of high temperature superconductors is of vital importance to provide a solid basis for testing these theories.
Inelastic neutron scattering (INS) is the traditional technique of choice to study magnetism, being able to detect magnetic fluctuations in the full Brillouin zone (BZ) 8,9 . Recently, Resonant Inelastic X-ray Scattering (RIXS) has emerged as a complementary technique to INS by detecting spin excitations in FePns close to the Γ point as summarized in Refs. [17][18][19][20][21][22] .
The detection of spin excitations is enabled in RIXS thanks to the spin-orbit coupling of the intermediate state mixing the quantum numbers L and S thereby activating a channel for the detection of magnetic excitations 23,24 . An important consideration when comparing RIXS with INS is the portion of BZ probed by the two techniques, close to the Γ point in the case of RIXS and at the AF wave-vector for the case of INS. Depending on the case this two positions in momentum space can be equivalent or not. Moreover, it is hard to estimate and compare the absolute weight of magnetic excitations in these two regions of BZ, but it is generally accepted that the intensity at the AF wave-vector is higher than close to the Γ point.
In this article, we use RIXS to systematically unveil the persistence and gradual hardening of the spin excitations in isovalently-doped BaFe 2 (As 1−x P x ) 2 across the phase diagram (see This apparent dichotomy implies that there is a transfer of magnetic spectral weight from localized to itinerant as imposed by sum rules relations. We argue that the balance between localized and itinerant states is the key to achieve SC and plays an important role for the physics of criticality in BaFe 2 (As 1−x P x ) 2 . In Fig. 4b,c, we depict the doping dependence of χ s directly extracted from the MRDF calculations as well as the RIXS spectra at (0.44,0). To better visualize the renormalization in energy of the magnetic excitations we also take the difference between selected doping levels and the parent compound and present the results in Fig. 4d for both theory and experiments. The agreement between theory and experiment is remarkable individuating the the magnetism and electronic structure of FePns. In previous works DFT-RPA has been employed to successfully describe the spin excitations in overdoped cuprates [39][40][41] . The failure of DFT-RPA to account for the spin excitations in BaFe 2 (As 1−x P x ) 2 implies that the FePns are not weakly correlated systems and cannot be compared to overdoped cuprates with reduced electronic correlations but rather need to be placed in the family of multiorbital correlated systems similarly to heavy fermion materials 16,39 .

X-Ray Emission Spectroscopy
To complement our measurements of the spin excitations and assess the local magnetism of BaFe 2 (As 1−x P x ) 2 , we employed X-ray Emission Spectroscopy (XES) -a classical technique that has been established as a sensitive probe of the local magnetic moment (µ bare ) 19,42-48 .
XES is sensitive to the local fluctuating magnetic moment and does not require a net magnetization or ordering, such as X-ray Magnetic Circular Dichroism (XMCD) or neutron diffraction, but detects directly the paramagnetic moment [44][45][46] . In this technique, a photon (hν =7140 eV) excites an Fe-1s core-electron into the continuum, creating a core-hole which is filled by a Fe-3p electron with the consequent emission of a photon (hν =7040- In Fig. 5a,b, we show XES spectra for BaFe 2 As 2 , and BaFe 2 (As 1−x P x ) 2 (x=0.52), and FeCrAs and the respective difference spectra. Clearly, a gradual decrease of µ bare is inferred from the difference spectra depicted in Fig. 5c. The values of µ bare , presented in Fig. 5d,e are continuously reduced, (for example µ bare = 1.0±0.1 for x =0.00, µ bare = 0.6 ± 0.1 for x =0.28, and µ bare = 0.4 ± 0.1 for x =0.52), but not fully quenched by doping, despite the complete disappearance of the ordered magnetic moment observed by neutron scattering 50,51 .
This evidence is remarkable in light of the constancy of the Fe oxidation state observed in XAS (see Supplementary Fig. 2) with isovalent doping driving the antiferromagnetically long range ordered BaFe 2 As 2 into a paramagnetic phase.
We compare the doping dependence of the local moment with the strength of the Fermi surface (FS) nesting at the antiferromagnetic wavevector. In Fig.5d, the theoretical data is the computed static susceptibility at the antiferromagnetic wavevector at doping x [we normalize ∆χ (x=0) =1]. Note that this comparison is only qualitative, and a self-consistent estimation of the static magnetic moment is computationally expensive. Similar doping dependence of the magnetic instability arising from the FS nesting and the observed local moment indicates an electronic mechanism of the magnetic ground state in this system.

DISCUSSION
Our XES results are puzzling and assuming spectral weight sum rules, there has to be spectral weight transfer from localized moments into spin excitations as probed by RIXS.
In XAS spectra for all samples at 15 degrees of incidence angle relative to the sample surface.
All the XAS spectra are reported in Supplementary Figure 3 and display the constancy of the iron oxidation state. There are small spectral differences at around 710 eV that are possibly due to a different covalency between the FeAs and FeP.
The RIXS spectrometer was set to a scattering angle of 130 degrees and the incidence angle on the samples surface was varied to change the in-plane momentum transferred (q // ) from (0, 0) to (0.44, 0) and from (0, 0) to (0.31, 0.31). All RIXS measurements in the present paper were recorded in grazing incidence configuration as depicted in Supplementary Figure   2. The zero energy loss of our RIXS spectra was determined by measuring spectra in σ polarization. The total energy resolution was measured employing the elastic scattering of carbon-filled acrylic tape and is around 110 meV.
RIXS spectra were normalized to unity and the main emission line was fitted according to Ref. [17][18][19][20]56 employing the following formulas: and In the first formula, the first part is a 2 nd order polynomial function describing the emission line at low energy loss, the second part is an exponential decay describing the emission line at high energy loss. The two behaviours are swapped into each other by the g γ term. The
Also accurate tight-binding models are available to reproduce the DFT band structure, consistent with Angle Resolved Photoemission Spectroscopy (ARPES) data (after including renormalization effects). However, P-doping effects on the As site, which does not change the carrier concentration, is neither trivial to accurately calculate within the DFT framework, nor within a simple rigid band shift technique which works reasonably well for electron and hole doping cases 58 . Since P atom is smaller in size compared to As atoms, P doping is expected to decrease the lattice volume. Indeed, both x-ray 52 , and neutron 51 diffraction analysis revealed that all three lattice constants as well as the pnictogen atomic coordinates in the unit cell (z P n ) and the pnictogen height from the Fe plane (h P n ) decrease monotonically with P doping. Among them, the c-axis lattice constant decreases drastically from ≈13 A at x = 0.00 to ≈12.4Å at Based on the aforementioned experimental observations on the band structure evolution with doping, we construct an effective five-orbital tight-binding (TB) model to reproduce the low-energy dispersion and FS topology across the entire phase diagram. The 2 Fe unit cell incorporates two Fe sublattices producing a 10 band model. We start with a five orbital TB model as derived in Ref. 57 for BaFe 2 As 2 . The intra-orbital dispersions are defined by where k is the crystal momentum and i = 1 − 5 is the orbital index.
i k is the momentum dependent part of the dispersion which arises from the nearest, next, and higher neighbour hoppings between the same orbitals, ∆ i is the corresponding on-site potentials, and F is the chemical potential. As the unit cell volume decreases monotonically with P doping, it is expected that the electron hopping amplitude increases monotonically with doping. We model this effect by a simple renormalization factor λ as i k → λ i k , where λ increases with doping. Similarly, due to the monotonic increase of the pnictogen coordinate and the height (z P n , h P n ), the on-site potential also changes. Interestingly, we find that an orbital dependent modification of the on-site potential is required to properly reproduce the experimental behaviour of the k z dispersion and FS changes. We set ∆ i → ∆ i + δ for      Fe-K β X-ray Emission Spectroscopy (XES) and difference spectra for