Probing spin correlations using angle-resolved photoemission in a coupled metallic/Mott insulator system

An intrinsically nonmagnetic spectroscopy is shown to have strong magnetic sensitivity in Kondo-coupled PdCrO2.


Supplementary Text
Text S1. Density functional theory.
Scalar-relativistic density functional theory (DFT) electronic structure calculations were performed using the full-potential FPLO code (31-33), version fplo18.00-52. For the exchangecorrelation potential, within the local density approximation, the parameterization of Perdew-Wang (34) was chosen. To obtain precise band structure, the calculations were carried out on a0020well converged mesh of 8000 k-points (20x20x20 mesh, 781 points in the irreducible wedge of the Brillouin zone). For all calculations, the experimental crystal structure was used (13).
From the converged calculation, a 10-band tight-binding model (TBM) based on Wannier functions was constructed in an energy window between -3 eV and 2 eV. For the TBM, all Cr 3d and Pd 4d orbitals were taken into account. This results in very good agreement with the DFT bands in the relevant energy window near the Fermi level (see fig. S1). The extracted tight binding integrals and on-site energies (see Table S1) were used as input parameters for the strong coupling theory (see Text S3).

Text S2. DFT + DMFT calculations: Spectral function
DFT + dynamical mean-field theory (DMFT) calculations for PdCrO 2 were carried for its experimental lattice structure (a=2.92 Å, c=18.09 Å) with a charge self-consistent framework (35, 36) combining the linearized augmented planewave band-structure code "Wien2k" (37) and the DMFT implementation provided by the library "TRIQS" (38, 39). Projective Wannier orbitals representing Cr 3 and Pd 4 states were constructed from Kohn-Sham bands in the energy range of [-5:11] eV. The rotationally-invariant on-site Coulomb repulsion vertex between Cr 3 orbitals was specified by the Slater parameter F 0 = 4.5 eV and Hund's rule coupling J H = 0.75 eV. We solved the DMFT quantum impurity problem employing the quasi-atomic Hubbard-I approximation (40). The double-counting correction was evaluated in the fully-localized limit Two ghost bands that have only weight for the unoccupied Cr 3d e g manifold have been removed for clarity. using the atomic occupancy 3 of the Cr 3 shell. Self-consistent DFT+DMFT calculations were converged to 0.05 mRy in the total energy. The resulting total k-resolved DFT+DMFT spectral function ( , ) is displayed in fig. S2. Within the Hubbard-I approximation the DMFT self-energy has no imaginary part, hence, there is also no lifetime broadening in our resulting ( , ). Apart from this, however, our calculated spectral function is in excellent agreement with that obtained in Ref.
(20) using a numerically exact quantum Monte Carlo method (one may notice a rather weak lifetime broadening for the occupied part of ( , ) shown in Fig. 3 of Ref. (20)).
The largest contribution of the Cr 3d character to the DFT+DMFT ( , ) is seen in the range from -2.5 to -2 eV, in good agreement with the location of diffuse spectral weight in the experimental on-resonance ARPES (Fig. 2B of the main text). There is almost no Cr contribution to the dispersive bands crossing E F , confirming that they are of dominant Pd 4d character. This finding is supported by our ARPES measurements: as well as a lack of spectral-weight enchancement of this band across the Cr L 2,3edge resonances (I MB in Fig. 2D of the main text), our high-resolution measurements indicate that its Fermi velocity is very similar in PdCrO 2 and the non-magnetic sister compound PdCoO 2 , and very fast in both ( fig. S3). This rules out any significant correlation-driven mass enhancement of these states.

Text S3. Strong coupling theory
The demonstration of a Mott insulating state of the CrO 2 layers justifies the strong-coupling theory for the spectral function of the PdCrO 2 utilised here. While we show a simplified version without the orbital degree of freedom in the main text, here we employ a more realistic fourorbital model represents the 3 2 − 2 orbital of Pd electrons on the th site, while represents the 3 orbitals of Cr electrons. = 1,2,3 labels the three low-energy states of the crystal field splitting | = 1〉 = | 3 2 − 2 〉, | = 2〉 = cos | 2 − 2 〉 + sin | 〉, | = 3〉 = sin | 〉 − cos | 〉 (2) In PdCrO 2 the weak interplane coupling renormalises the Fermi velocity by at most 6% as compared to PdCoO 2 , although a slight lattice expansion for PdCrO 2 vs. PdCoO 2 means that the actual degree of correlation-driven mass enhancement is likely even smaller than this value. This justifies the treatment of the Pd electrons as uncorrelated in the intertwined spin-charge model discussed in the main text.
with tan~0.693, which are schematically drawn in fig. S3. Pd and Cr respectively form triangular lattices and are stacked alternately. The hopping parameters are estimated from the 10-orbital (Pd 4 and Cr 3 electrons) model constructed from the first-principles calculations (see Text S1), and are summarized in Table S1. In reality, the metallic Pd band of the 3 2 − 2 orbital is substantially hybridized with and 2 − 2 orbitals. While the multi-orbital treatment is necessary for orbitally-resolved properties (such as the Berry curvature [arXiv:1811.03105]), it is not crucial for reproducing the parabolic energy spectrum and so we neglected the detailed orbital components of Pd electrons for simplicity. The derivation of the Kondo Hamiltonian and the associated spectral formula is streightforwardly applicable to the case where the Pd electrons have the orbital indices as well.
We explicitly include the onsite repulsion and the Hund coupling for the Cr electrons as to capture the correlated nature of the 3 electrons. Here = ∑ † and is the localized spin of Cr.
where is determined such that terms changing the number of doublon, holon, or 2 = ( + 1) do not appear in the effective Hamiltonian. The Schrieffer-Wolff transformation thus `eliminates' the high energy charge (doublon-holon) excitation of the Cr atoms. The upper and lower Hubbard bands, the latter of which is visible in experiment ( Fig. 2(B,C) of the main text) is thus missing in the Kondo-Heisenberg model. However, all excitations involving the spin degree of freedom are kept in the model after the transformation. We consider a perturbative expansion from the atomic limit and construct order by order: We expand = (1) + (2) + ⋯ and compare each order, where hopping terms are allocated to the first order and interacting terms the zeroth order. From this condition, the first-order term (1) must satisfy where | ⟩ is the (macroscopically-degenerate) ground state (no doublons/holons, = 3/2 on each site) of int and eff = + 2 . We obtain the second-order effective Hamiltonian (up to constants) from this relation as The obtained form is identical to that of the simplified version (Equation 2 of the main text) but with = 3/2 rather than = 1/2 . The last term can be thought as the Kondo coupling between the local Cr spin and neighbouring Pd electrons. Representative values are determined from the DFT calculations, and are listed in Table S2.
These values of superexchange interactions between Cr = 3/2 spins are in good agreement with those evaluated from the converged DFT+DMFT results with the linear-response theory of Ref. (41) for several first correlation shells ( fig. S5). The obtained values are listed in Supplementary Table S3. and (−2,0,1) respectively.

Spectral function
The spectral function of Cr electrons is represented as where | 0 ( ) ⟩ is the eigenstate of the original Hamiltonian with electrons and eigenenergy   fig. S4.
with ( ) being the Heisenberg representation (for the effective Hamiltonian) of an operator . Namely, the low-energy excitation of the Cr electrons is described by a simultaneous disturbance on the metallic layer and the localized spin.
Since the coupling between the metallic and insulating layer is small (~| | 2 eff ⁄ ), we can decouple the expectation value 〈 † 〉~ 〈 〉 〈 † 〉 in the leading-order evaluation. This treatment is justified when the vertex correction (in terms of the diagrammatic expansion about the inter-layer coupling) can be neglected. The correlation function can then be approximated as where we neglect the off-diagonal element 〈 + , ( ) + ′ , ′ † 〉 which can exist in the ordered phase. Equation (3) in the main text is obtained by considering the zero temperature limit of the Fourier-transform of the expression for Cr eff ( , ).
The spectral intensity of the back-folded band with the momentum shift of ± can be approximated as Cr ± ( ) = ∑ 8 2 | ± | 2 9 eff 2 (15) (fits to the calculated Cr spectral function are plotted as a function of energy in Fig. 3A of the main text (solid line)). This form of the intensity is substantially different from that of the backfolded band of Pd electrons due to the modified potential.
If we consider a bilayer case for simplicity, the 'band folding' model Pd Hamiltonian, with the modified potential due to the 120-degree structure, can be written in a 2 × 2 form The spectral intensity of the back-folded band with the momentum shift of ± is obtained as which rapidly decays as Pd ( )~2 | /( − + )| 2 as | − + | becomes large. We plot this form of intensity in Fig. 3A of the main text (dashed line), with ∆= 20 meV, consistent with the experimental gap 2∆ ~40 meV estimated from the breakdown field reported in Ref. 16 as ℏω c~Δ 2 / . We stress that, for such a "conventional" back-folding, the rapid suppression of spectral weight away from the magnetic Brillouin zone boundary is therefore intrinsic. In contrast, in the intertwined spin-charge model, the reconstructed weight in the Cr spectral function is approximately constant, varying only due to momentum-dependent variations of the inter-layer coupling term g k+Q . Our calculations ( fig. S6) show how this is sensitive to details of the calculation. Including only nearest-neighbour coupling, the spectral weight slightly decreases towards the Fermi level. Including next nearest-neighbour coupling, the spectral weight increases towards the Fermi level. In all cases, the variation in spectral weight is less than a factor of 2 over an energy range of more than 700 meV below the Fermi level, entirely consistent with our experimental measurements shown in Fig. 3A of the main text and Fig. S6. Throughout, we have shown the relative intensity variations between different models and the experimental data by normalising to the intensity at -0.7 eV binding energy; equivalent conclusions are drawn if normalising directly by the main band intensity, as shown in Fig. S7. Although varying ARPES matrix elements cause some changes in the binding energy dependence of the reconstructed weight intensity for the different measurement conditions, the measured spectral weight never varies by more than a factor of two over the 700meV energy range. This is in sharp contrast to the prediction of the simple 'band folding' model. While binding energy independent 'shadow features' have previously been observed in materials with superperiodic structures or structural distortions (42), such underlying origins would be inconsistent with the Cr character of the reconstructed weight that we observe. (C) Cr I RW predicted by the intertwined spin-charge model for various combinations of nearest neighbour (nn) and next-nearest neighbour (nnn) hopping parameters. Small parameter-dependent quantitative variations are observed, although the overall binding energy dependence of the reconstructed weight intensity remains weak for all parameters, again in clear contrast to the 'band folding' model.