Kondo screening in Co adatoms with full Coulomb interaction

Using a numerically exact first-principles many-body approach, we revisit the"prototypical"Kondo case of a cobalt impurity on copper. Even though this is considered a well understood example of the Kondo effect, we reveal an unexpectedly strong dependence of the screening properties on the parametrization of the local Coulomb tensor. As a consequence, the Kondo temperature can vary by orders of magnitude depending on the complexity of the parametrization of the electron-electron interaction. Further, we challenge the established picture of a spin-$1$ moment involving two cobalt $d$-orbitals only, as orbital-mixing interaction terms boost the contribution of the remainder of the $d$-shell.


I. INTRODUCTION
The Kondo effect arises when a local magnetic moment is quantum mechanically screened by the conduction electrons of a metallic host.Explained by Jun Kondo in the 1960s 1 , this phenomenon has been extensively studied thereafter within Anderson's poor man's approach and Wilson's renormalization group 2 .As a direct consequence of the screening of the impurity magnetic moment, the spin susceptibility undergoes a crossover from a Curie-Weiss to a Pauli behaviour upon lowering the temperature.At the same time, the Abrikosov-Suhl-Kondo resonance [2][3][4] emerges in the electronic spectral function at the Fermi level.Magnetic response functions and electron transport are therefore suitable probes of the Kondo effect.Despite its well-defined characterization, the signatures of the Kondo effect emerge at energy scales of the order of the Kondo temperature T K , which is often of the order of a few Kelvin, making the theoretical description of realistic Kondo systems intrinsically hard.Further, the Fermi-liquid properties emerging below the Kondo temperature T K are typically reached via smooth crossovers rather than with sharp transitions, complicating also the experimental detection.
One case of Kondo effect considered to be simple and relatively well understood is that of a Co single impurity on a metallic substrate, such as Cu [5][6][7][8] , Au 6 , and Ag 6,9 .In particular, scanning tunneling spectroscopy (STM) has revealed how Co adatoms on Cu hosts display sharp peaks or Fano-like resonances at zero bias [5][6][7][8]10 , which are commonly interpreted as a clear experimental signature of the Kondo screening, although the origin of these features has been recently challenged 11 . Exerimental estimates of the Kondo scale yield, e.g., T K ≈ 88 K and 54 K for Co on Cu(001) and Cu(111), respectively 5,6 .
However, even in the case of a single impurity, for tran-sition metal adatoms the theoretical description of the Kondo effect is difficult, since the whole d-shell is likely to play a role in the screening.So far, the theoretical understanding of single Co impurities on Cu [12][13][14][15] stresses the main role played by two of the five Co-d orbitals.In the case of Co/Cu(001) -on which we shall focus below -the d xy and d z 2 orbitals are Kondo active, in the sense that in the ground state of the Co impurity, they are halffilled and carry a magnetic moment.Due to the different symmetry, for Co/Cu(111) the d z 2 orbital is instead fully occupied, and the magnetic moment arises from one of the two doublets with E symmetry 12,15 .In general, different crystalline environments determine variations in the local electronic structure of the impurity and lead to drastically different Kondo resonance line shapes observed in STM experiments 8,12 .Furthermore, the manybody nature of the Kondo effects manifests itself also in a strong dependence of T K on the occupation of the Co 3d shell 6,8,13 .This also means that the hybridization and the charge transfer between the impurity and the substrate play an important role.This is reflected in a strong dependence of T K on, e.g., the adatom adsorption distance 15,16 , in agreement with the experiments 7,8 .In general, the Kondo scale depends exponentially on the parameters of the theoretical model, making reliable estimates of T K extremely hard.
For the same reason, it is also difficult to exactly pinpoint the details of the physical processes underlying the Kondo screening in these systems.Interestingly, theoretical calculations indicate that the spin state of Co is S = 1 on both the Cu(001) and Cu(111) surfaces 15 .However, evidence for very different Kondo scales for the Kondo-active orbitals of Co/Cu(001) 13,15 suggest an underscreened (or possibly a two-stage 17 ) Kondo effect to take place, while a single T K is expected for the magnetic doublet of Co/Cu(111), although the degener- acy could be lifted by, e.g., spin-orbit coupling 15 .On the other hand, Nevidomskyy and Coleman 18 showed that, in the case of a multi-orbital impurity, the stabilization of an impurity high-spin state due to Hund's coupling leads to a strong reduction of the Kondo coupling, and consequently of T K , with respect to the spin S = 1/2 case 19,20 .Robust numerical evidence that the Nevidomskyy-Coleman scenario is realized in idealized model systems comes, e.g., from Ref. 21 .This seems, however, at odds with the relatively high estimates of T K for these systems emerging from experiments [5][6][7] .Hence, the question is whether or not, or under which conditions, the Kondo screening of Co on a Cu substrate can be described this way upon cooling.
We identify two key players which may affect the mechanism of the Kondo screening, i.e., multi-orbital correlation effects arising from the full treatment of the Co 3d shell, rather than restricting the description to the Kondo-active orbitals only, and the approximation of the form the Coulomb interaction.Using a combination of density functional theory (DFT) and numerically exact quantum Monte Carlo (QMC) we analyze the many-body processes leading to the formation and the screening of the local moment on a Co impurity on Cu(001) in its full realistic complexity.We provide a comparative analysis of the role of the parametrization of the Coulomb interaction, which is so-far scarcely investigated in a systematic way.In particular, we take into account the full Coulomb tensor in the whole Co 3d multiplet, hitherto either simplified 13,15 or included only at high tempera-tures 14,22 , and we push our calculations down to temperatures which are relevant to the Kondo screening.
The paper is organized as follows: In Sec.II we provide the details of the ab-initio and many-body calculations for Co/Cu(001).In Sec.III we discuss the possible Kondo scenarios, and in Sec.IV we analyze the screening properties, providing evidence which supports the important role played by the approximations of the Coulomb tensor.Finally, Sec.V contains a discussion of our results in light of previous studies in the literature, as well as our conclusions.

A. DFT++
Here we investigate the correlation effects of a single Co adatom on a Cu(001) surface using a combination of DFT within the local density approximation (LDA) and the numerical solution of an Anderson impurity model (AIM) with realistic parameters.This approach is commonly referred to as DFT++ in the literature 23,24 .
The DFT calculations have been performed with the Vienna ab-initio simulation program (VASP) 25,26 using the projector augmented plane wave (PAW) basis set.We modeled the Cu(001) surface as a 4×4 slab consisting of five Cu layers using the experimental 27 lattice constant of 3.615 Å.The Co adatom was placed in the fourfoldhollow position (see Fig. 1) at an adsorption distance of 1.52 Å with respect to the first Cu(001) layer, which we identified in one of our earlier works 16 to be the energetically favored distance, in agreement with previous literature 12 .We used a k-mesh centered around the Γpoint of size 100 × 100 × 1 k-points in order to achieve a sufficiently accurate description of our Cu(001) substrate.This will be necessary for the parametrization of the AIM, especially at low temperatures (this important aspect is discussed in Appendix A in more detail).
With the combination of DFT and an AIM, we can take into account the correlation effects on the Co atom explicitly as well as the realistic complexity of its hybridization with the Cu substrate.The Hamiltonian of the AIM reads where ĉ † νσ (ĉ νσ ) denotes the creation (annihilation) operators for an electron with spin σ in the νth bath state (in this work, the Cu surface) with energy ν , whereas d † iσ ( diσ ) denotes the corresponding operators for the ith localized 3d orbital of the impurity (in this work, the Co 3d shell) with energy i .The bath and impurity electrons are coupled via the hybridization V νi .For QMC techniques, it is convenient reformulate the AIM (1) in the action formalism, and integrate out the degrees of freedom of the bath to obtain a retarded (i.e., frequencydependent) hybridization function ∆ iσ (ω) which effectively embeds the impurity into the substrate.Our results are compatible with those found in the literature 13 .The hybridization function is then transformed into the Matsubara representation ∆ iσ (ω) → ∆ iσ (ıω n ) for QMC sampling.Finally, the tensor describes the local Coulomb interaction (where we dropped the spin indices, for simplicity) as introduced by Slater 28 , with ψ α (α = i, j, k, l) being in general any atom-centered basis function and e 2 |r−r | the long-range Coulomb potential.

B. Coulomb tensor
The last term of Eq. ( 1) describes the local interaction within the impurity 3d shell.The full Coulomb interaction U ijkl is in general a four-index tensor, which, in the language of second quantization, corresponds to different combinations of the creation and annihilation operators of the two-body interaction.However, due to the extreme numerical complexity required to take into account all possible four-fermion terms, it is common practice so far to consider approximate interaction schemes.Therefore, most previously published results have been obtained neglecting -in a non-systematic and uncontrolled way -parts of the Coulomb tensor.With the advent of continuous-time quantum Monte Carlo (CT-QMC) methods (see Ref. 29 for a review), it has become possible to treat the full Coulomb interaction without approximations.There are already indications in the literature 22 that the structure of the full Coulomb interaction is important to describe the physics of Co/Cu(001).We will show that different parametrizations of the Coulomb interaction also give rise to substantially dissimilar Kondo screening properties.
Below we describe the properties of U ijkl in different approximation schemes.Within the simplest parametrization, one retains only the "density-density" terms, i.e., those in which the four operators are contracted in pairs of number operators niσ = d † iσ diσ .Within the density-density approximation, the Coulomb tensor in Eq. (1) reduces to The parameters U ii and U i =j derive from U ijij , and denote the intra-and inter-orbital (direct) interactions, while J ij = U i =j ijji denotes the Hund's exchange coupling.Including also the missing two-body scattering terms, which describe "spin-flip" or "pair-hopping" processes between different orbitals, gives rise to the so-called "Kanamori" parametrization, of the form which has the important consequence of restoring the rotational invariance of the Coulomb interaction.Finally, the "full Coulomb" interaction, given by the generic form contains all possible terms allowed on the 3d shell, without restrictions.In the case of a spherically symmetric atom, these terms can be described in terms of the Slater radial integrals 28,30 F 0 , F 2 , and F 4 .With a sphericallysymmetric Coulomb tensor, one has the advantage of excluding sources of differences associated with specificities of the Cu(001) substrate, at the same time allowing us to reduce the number of interaction parameters to two: U = F 0 and J = 1 14 (F 2 + F 4 ).For instance, the intra-orbital Hubbard repulsion becomes independent of the orbital index i and is given by the relation ).The different angular dependence of the five d-orbitals results in four different Hund's couplings J ij , which can all be expressed in terms of F 2 and F 4 , so that U ij = U ii − J ij (see, e.g.Refs. 31,32for a detailed discussion).

C. Details of the Co/Cu(001) calculations
We solve the AIM (1) by using the numerically exact CT-QMC method as implemented in w2dynamics 33,34 .With the choices of U = 4.0 eV, J = 0.9 eV, and the ratio F 4 /F 2 = 0.625, which completely determine the Coulomb tensor, the values we use in this study are very similar (although spherically symmetric) to those calculated for Co/Cu(001) from first principles by Jacob within the constrained random phase approximation 13 .For the purpose of showing how significant the differences between the results obtained within various interaction schemes can be, we compare the magnetic properties of Co/Cu(001) obtained by solving the impurity problem with the Coulomb interaction ĤC of Eq. ( 5), as well as with its density-density and Kanamori approximations of Eqs. ( 3) and ( 4), respectively.We will show that different approximations of the Coulomb tensor lead to different physical pictures.In particular, the lowest temperature reached here for Co/Cu(001) in the scope of the full Coulomb interaction is T 33 K, which is below the experimental estimates of T K for this system.To the best of our knowledge, it is the first time that the analysis of a single Co adatom on Cu has been pushed to such low temperature in the framework of a five-orbital AIM with the full Coulomb interaction.
When interfacing many-body and ab-initio calculations, as within the DFT++ scheme, one should also be aware of the so-called double-counting problem, which one encounters because part of the correlation energy (in this case on the Co 3d shell) is already taken into account within DFT.Usually, one approximates the doublecounting value from the fully localized limit (FLL) 35 or  18 can be realized at very low TK .As inter-orbital spin correlations are weaker and charge fluctuations are enhanced, an underscreened (or possibly two-stage) Kondo effect may take place at substantially higher TK .A collection of S = 1/2 replicas would be favored at JH = 0, but could also emerge due to the competition between all possible generalized exchange interactions (e.g., of the form U ijjk ) within the Co 3d shell.More complex SU(N > 2) spin-orbital Kondo scenarios, depending on the degree of orbital degeneracy, would also lead to an enhancement of the Kondo scale.
the around mean-field (AMF) 36 methods.Here we follow an alternative procedure, and choose the double counting in order to fix the Co 3d occupation to n d = iσ n iσ = 8 electrons, instead.One reason behind this choice is that the system has been investigated in several theoretical studies in an STM-like setup 13,15,37 , where it is assumed that Co on Cu(001) has an S = 1 spin state with an overall Co 3d occupation of roughly n d = 8 electrons.This was also confirmed by correlated wave-function-based calculations where a Co/Cu n cluster is embedded in a periodic potential 12 .Under the effect of the substrate crystal field, in the atomic ground state, the Co d x 2 −y 2 and the (d xz , d yz ) doublet are completely full while the d xy and d z 2 orbitals are both half-filled.In this situation, S = 1 high-spin configurations are expected to be locally dominant, which calls for a systematic analysis of the role of the Hund's coupling within the different approximations of the Coulomb tensor.However, we will also discuss deviations from integer filling of the Co 3d shell, as they are expected to influence the screening properties of Co/Cu(001) 13 .

III. POSSIBLE KONDO SCENARIOS
The goal of this section is to determine how the Kondo screening mechanism can be influenced by the parametrization of the local Coulomb repulsion on the Co impurity.To this end, we are going to analyze in particular the finite-temperature spin and charge response functions, calculated at the Co site.We compare the three interaction schemes discussed in Sec.II B (i.e., densitydensity, Kanamori, and full Coulomb), especially focusing on the Co d xy and d z 2 orbitals, which are identified as the Kondo-active orbitals in the literature 13,15 (note the different orientation of the xy plane here compared with these works).However, we will claim that more realistic descriptions of the Coulomb tensor favor a scenario in which also the other 3d orbitals play an important role in the screening of the Co local moment.
The scheme presented in Fig. 2 anticipates the main results of the present paper.The ground state electronic configuration of the Co 3d shell in the Cu(001) crystal field is show in Fig. 2(a).In the Co with n d = 8.0, a high-spin state is always realized for temperatures above the Kondo regime.We find a link between the form of the Coulomb interaction, which strongly affects spin-and charge fluctuations, as represented schematically in Fig. 2(b), and the possible mechanism behind the screening of the Co spin, indicated in Fig. 2(c).In the simplest approximation scheme, i.e., the approximation of Eq. ( 3), in which only the density-density part of the local Coulomb interaction is taken into account, the (d xz , d yz , d x 2 −y 2 ) subspace is completely filled, and hence inert.Due to the strong Hund's coupling within the Kondo-active subspace, the Co impurity is locked into an S = 1 state down to a few K, when it will eventually be screened by the conduction electrons of the Cu surface, thus realizing the Nevidomskyy-Coleman scenario of the suppression of T K for an S = 1 Kondo impurity.We increase the complexity of the Coulomb tensor, by including interaction terms beyond densitydensity in the Kanamori (first) and the full Coulomb (later) parametrizations.We observe a progressive breaking of the S = 1 high-spin state, and different screening mechanisms become favorable over the one proposed in the Nevidomskyy-Coleman scenario.We rationalize this effect in terms of two key players: i) the enhancement of charge fluctuations within the whole Co 3d multiplet, and ii) the frustration of the spin correlations due to the competition between all generalized exchange interactions in the Coulomb tensor, e.g., of the form U ijjk , which include both the Hund's coupling (i = k) and additional processes beyond the density-density approximation (i = k).
It is interesting to speculate on the suitable screening mechanisms which could replace the Nevidomskyy-Coleman scenario for Co/Cu systems, in order to look for their characteristics in our numerical analysis.One possibility is the underscreened Kondo effect, where the Co spin is only partially screened by the substrate.Depending on how many modes of the host effectively couple to the impurity, a Noziéres Fermi liquid can be recovered at lower T by screening the remaining spin (thus realizing a two-stage Kondo effect).In the regime where the charge fluctuations become dominant, the d xy and d z 2 orbitals may also behave as a pair of S = 1/2 replicas, which are screened at possibly very different T K s.Moreover, depending on the degree of orbital degeneracy of the 3d multiplet, an SU(4) Kondo effect could also take place.The latter may be relevant for the Co/Cu(111) case, where the Co magnetic state is actually a doublet 15 .The increased symmetry, from an SU(2) spin-Kondo to an SU(4) spin-orbital Kondo -or even an SU(N) symmetry, involving also the rest of the 3d multiplet -is generally expected to result in a single enhanced Kondo scale 38 .All the above mechanisms would be compatible with the relatively high T K ∼ 50 − 100 K estimated by transport experiments [5][6][7] .
While the general role of the Coulomb interaction emerges clearly from our calculations, a precise estimate of T K and the identification of the Kondo mechanism responsible for the screening for each parametrization of the Coulomb tensor remains elusive.This is mostly due to the difficulty of observing typical Fermi liquid temperature scaling within our methodology.

A. Spin correlations and effective local moment
In order to investigate the screening of the impurity magnetic moment we sample the spin-spin response function in imaginary time within CT-QMC: where i and j denote the Co 3d impurity orbitals, Ŝz i is the local spin operator on orbital i, and g is the electron spin gyromagnetic factor.The static (i.e., ω = 0) spin susceptibility is obtained via integration of the diagonal elements of Eq. ( 6), as where β is the inverse temperature.For a Kondo impurity, the static spin susceptibility follows a Curie-Weiss behavior χ(T ) ∝ 1/T in the local moment regime well above T K .As the moment is screened by the conduction electrons, the spin susceptibility has a crossover to a Pauli behavior due to the onset of a Fermi liquid (FL) regime: χ −1 (T ) ∝ T +T F L , with the characteristic coherence temperature T F L corresponding to T K in the case of a single impurity 39,40 .
In Fig. 3 we compare the spin susceptibility of the d xy and d z 2 orbitals obtained for all interaction parametrizations.In the corresponding inset we also plot T χ ii (T ), as it is customarily done in order to represent a Curie-Weiss susceptibility as a constant and a Pauli susceptibility as linearly vanishing.This allows to highlight the differences observed with the three interaction schemes.Within the density-density approximation, we obtain an almost perfect 1/T behaviour of the susceptibilty (in the main panels, and a plateau in the insets) for both orbitals, indicating a Curie-Weiss behavior in the full range of temperatures of our calculations.Consequently, we can infer that the upper bound for the Kondo temperature within the density-density approximation is substantially lower than 30 K, i.e., it is likely of O(1) K. Instead, the Kanamori and full Coulomb parametrizations yield clear signatures of Kondo screening in the same temperature window.The Kanamori coherence scale seems to be still quite low, and at about 30 K the crossover from a residual entropy to a fully screened moment is indeed far from being complete.Within the full Coulomb parametrization, we observe a pronounced departure from a constant T χ(T ).With the exact (though still spherical) local Coulomb tensor, the Kondo temperature is hence about two orders of magnitudes larger than within the density-density approximation, which was extensively used in the past.We also note that a clear linear behavior of T χ(T ) is observed well above T ∼ 100 K for the d z 2 orbital (yet not for d xy one) only when the full Coulomb interaction is considered.This observation can also be justified by looking at the evolution of the electron occupation of the two orbitals, as we discuss in Sec.IV B. Further insight in the different screening processes activated by the Coulomb interaction can be obtained by inspecting two special values of the impurity spin susceptibility in imaginary time: χ(τ = 0) and χ(τ = β/2).At τ = 0, it corresponds to the (square of the) bare magnetic moment, sometimes also called the unscreened paramagnetic moment.It indicates the tendency of the Co impurity to build up a quantum magnetic moment at short time scales.Instead, its value at τ = β/2 can be associated to a magnetic moment at asymptotically long times, and hence it provides information on the effectiveness of the dynamical screening due to quantum fluctuations 41 .These two quantities are helpful to visualize the different screening properties within the three interaction schemes and allow us to understand which two-body processes are decisive for the Kondo screening.
In a correlated system we expect a strong contribution from the orbital off-diagonal components of the spin susceptibility, and in particular, in the case under study they are equally important as the diagonal ones.We inspect the screening properties by looking at the total (unscreened and screened respectively) "effective" spin moment S eff .This involves all components of χ ij (τ ) and takes into account the difference between the quantum nature of the spin degrees of freedom of the three parametrizations of the Coulomb interaction.Within the density-density approximation we describe an Ising spin, so that the (instantaneous, i.e., τ = 0) magnetic moment is given by Screened magnetic moment m 2 (β/2) within the whole 3d shell (filled symbols) and restricted to the (dxy, d z 2 ) subspace (solid line).The dashed lines separates intra-and inter-orbital contributions within the subspace.Due to the increasing contribution of orbitals outside the subspace, a two-Kondo-active orbital description of the magnetic moment becomes unsatisfactory for realistic Coulomb interactions.
Instead, since the Kanamori and the full Coulomb parametrizations preserve the spin SU(2) rotational invariance of the Coulomb tensor, the magnetic moment is given by We can hence define m 2 = ξ ij χ ij (τ = 0), where ξ = 3, except for the density-density case in which ξ = 1, and the indices i and j in the summation run over either all Co 3d orbitals, or over a subset thereof, as necessary.The natural generalization at finite imaginary time is therefore which allow us to extract the effective spin S eff (τ ) from the relation m 2 = g 2 S 2 eff for density-density (Ising spin), or m 2 = g 2 S eff (S eff + 1) for Kanamori and full Coulomb interactions (Heisenberg spin) 41 .
The empty symbols in the three upper panels of Fig. 4 show the unscreened (i.e., instantaneous) effective spin S eff (τ = 0), including the intra-and inter-orbital contributions from the whole Co 3d shell.For all interaction parametrizations we get an instantaneous paramagnetic spin moment S eff > 0.9, in excellent agreement with the value S = 1 expected in the high-spin configuration, and with the literature 13 , which remains perfectly constant in the whole range of temperatures considered here.The screened effective moment S eff (τ = β/2) at each temperature is suppressed with respect to its τ = 0 counterpart by quantum fluctuations.Within the densitydensity approximation, we observe a sizable effective moment S eff (β/2) ≈ 0.8 down to the lowest temperature investigated.This mirrors the information obtained by the analysis of the static spin susceptibility, and substantially rules out any temperature-dependent (i.e., Kondo)  screening of the local moment in this temperature window within the density-density approximation.In contrast, within both the Kanamori and the full Coulomb parametrizations we observe the pronounced screening of the "long-time" local moment, which is considerably stronger than what observed within the density-density approximation.At the same time, a clear temperature dependence of S eff indicates a strong ability of the environment to Kondo-screen the Co impurity spin.Therefore, even at integer filling of the Co 3d shell (n d = 8), the local quantum fluctuations described by more complete parametrizations of the Coulomb interaction disgregate the high-spin state already in the high-temperature regime, and favor the onset of Kondo screening.Instead, this does not happen in the density-density case, for which the Nevidomskyy-Coleman scenario of a strong suppression of T K for a spin S = 1 is fully realized.
It is interesting to analyze the orbital character of the impurity magnetic moment.We look at the (screened) partial magnetic moment (which is an additive quantity, unlike S eff ) obtained by restricting the double sum over i and j in the definition of m 2 to the (d xy , d z 2 ) subset of orbitals.We also calculate m 2 intra (i = j) and m 2 inter (i = j) within the subspace, and compare them to the value for the whole 3d shell.As shown in the lower panels of Fig. 4, within the density-density approximation, S eff (β/2) is mostly determined by the (d xy , d z 2 ) subspace, whereas in the case of the Kanamori and full Coulomb parametrizations there is an increasingly large contribution from the d xz , d yz , and d x 2 −y 2 orbitals.In the full Coulomb case, the (d xy , d z 2 ) subspace contributes to approximatively half of the effective moment.This demonstrates that a two-Kondo-active orbitals description of the system is no longer valid when a realistic Coulomb interaction is taken into account.The intra-and inter-orbital contributions to the local moment within the (d xy , d z 2 ) subspace are similar to each other for all three parametrizations, but both are strongly suppressed by introducing interaction terms beyond the density-density approximation.As we discuss in Sec.IV B, this can be understood by considering the charge distribution within the Co 3d multiplet in relation to the spin-locking tendency of the Hund's coupling.

B. Charge fluctuations and decoupling
The full Coulomb tensor (even if here it still assumes a spherical environment) represents the reference point in our comparative analysis, as it gives the most coherent of all the results and the largest Kondo temperature, meaning the closest to the experiments.In order to ascertain the origin of the physical differences between the full Coulomb and the two other approximate schemes we consider the generalized double occupations niσ njσ for parallel (σ = σ) and anti-parallel (σ = σ) spin orientations.The numerical data representative of the high-temperature regime (at T = 232 K) are collected in Fig. 5 and illustrated by a set of matrix heatmaps.For σ = σ , the diagonal elements correspond to the spin-and orbital-resolved occupations niσ .Note that all quantities are symmetrized over both spin (σ ↔ σ ) and orbital (i ↔ j) indices.Within the density-density approximation, both the d xy and d z 2 orbitals are close to half-filling (i.e., niσ = 0.5 electrons) and have well defined local moments.All the other Co 3d orbitals are almost full.Moreover, within the (d xy , d z 2 ) subspace, niσ njσ niσ njσ , for i = j, which marks the clear tendency towards a S = 1 high-spin configuration favored by the Hund's coupling J H = U i =j ijji .Within this picture, which is very similar to the atomic ground state configuration 12,15 , not only can one identify d xy and d z 2 as the Kondo-active orbitals, but one could naïvely expect the physics to be described to a good degree of approximation by a two-orbital AIM, as also assumed in previous literature 15 .
The situation is substantially overthrown in the case of the Kanamori and full Coulomb parametrizations.In fact, by progressively including more interaction terms beyond the density-density approxiamtion, i.e., moving from left to right in Fig. 5, two trends emerge clearly: (i) There is a significant charge redistribution within the Co 3d shell.In particular niσ in the (d xy , d z 2 ) subspace increases as (0.57, 0.60) → (0.59, 0.64) → (0.64, 0.77), resulting in the suppression of the local moment of the Kondo-active subspace observed in Fig. 4. (ii) The interorbital (i = j) double occupations for parallel and antiparallel spin orientations become similar, i.e., niσ njσ niσ njσ for all pairs of orbitals.As a consequence, the tendency towards a high-spin state of the (d xy , d z 2 ) pair is substantially weakened.At the same time, the d xy and d z 2 orbitals still possess the two largest local moments of the entire multiplet, so that they supposedly maintain a prominent role in the Kondo screening process, but with important contributions to the physics coming from the other orbitals.The results are in complete agreement with the conclusions of the spin susceptibility analysis.
The considerations above can also be better understood by explicitly calculating the charge fluctuations, defined as C ij ≡ δn i δn j = ni nj − ni nj , where we introduced the symbol ni = ni↑ + ni↓ .In the full Coulomb parametrization, we observe a significant enhancement of charge fluctuations with respect to the density-density approximation.By separating the contributions of C intra = i C ii and C inter = i =j C ij , analogously as for the spin susceptibility, we observe that both increase (in modulus, since C inter are negative) as well as their sum.We also note that the strongest relative enhancement obtained by increasing the complexity of the Coulomb interaction is observed in the inter-orbital charge fluctuations within the (d xy , d z 2 ) subspace.The enhancement of charge fluctuation is the hallmark of increased metallicity for the (Kanamori and) full Coulomb parametrization(s), as was also previously reported in model studies of multi-orbital impurity problems 42 .and it drives the Co impurity away from the regime dominated by the Hund coupling 43,44 .The two-body mixing terms, involving combinations of three (e.g., U iijk ) or even four (U ijkl ) different orbital indices, included within the full Coulomb parametrization are highly effective in reducing the "orbital rigidity" and eventually yield a solution which is well described neither by a single S = 1 Kondo effect 18 nor by two independently screened S = 1/2 spins 13,15 .
Interestingly, the temperature dependence of both the charge distribution within the Co 3d shell and the charge fluctuations is negligible with respect to the changes observed between different interaction schemes, so that the above picture is valid in the whole range 300 − 30 K, and probably still holds below that.

C. Spectral signatures of the Kondo effect
Useful insight can also be obtained by looking at the orbital-resolved spectral function of the Co 3d shell.While one may estimate T K (or at least an apparent T K at T = 0) from spectral features such as the width of the resonance 13,15 , we will refrain from doing so.Since our spectral functions are obtained with a numerical analytic continuation procedure (maximum entropy method), we only take them as qualitative indications of the redistribution of the spectral weight.First, we consider results within the density-density approximation, which are shown in Fig. 6. functions of both orbitals.The analysis of the charge redistribution within the 3d shell (upper panels of Fig. 6) shows that, upon adding the extra ∆n d = 0.2 electrons, the occupation of the Kondo-active orbitals increases as (0.57, 0.60) → (0.63, 0.71).However, part of the charge accumulating in the subspace comes from the rest of the Co 3d shell.This is indicated by the red arrow in Fig. 6, where the extra block denotes the average occupation of the (d yz , d xz , d x 2 −y2 ) subspace.Such a charge redistribution is detrimental to the stabilization of the high-spin state, which one realizes by comparing the inter-orbital double occupations niσ njσ and niσ njσ (connected by red dots and a line in the upper panels of Fig. 6).The effect of charging is qualitatively analogous to, yet not as strong as, what we observed by comparing the densitydensity and full Coulomb parametrizations at n d = 8.0 in Fig. 5. Hence, both scenarios are compatible with an enhancement of the Kondo scale.
A clear signature of the Kondo effect is indeed observed already at integer filling in the full Coulomb spectral function, which is shown in Fig. 7 for different temperatures.A clear resonance close to the Fermi level is observed for both the d xy and the d z 2 orbitals.The resonance is already present at T ≈ 100 K, but it gets progressively closer to the Fermi level and its width decreases as the temperature is lowered (see side panels of Fig. 7).Interestingly, within this interaction scheme, a low-energy resonance develops also in the (d xz , d yz ) doublet.This feature is almost completely absent within the density-density approximation, and it can be regarded as a further indication that a purely (d xy , d z 2 ) description of the Kondo effect is not adequate, when accounting for a realistic Coulomb interaction in Co/Cu(001).Similar resonances are also evident in the spectral functions obtained away from integer filling, where the role of the other three orbitals is further enhanced.

V. DISCUSSION AND CONCLUSIONS
In this work we investigate the Kondo screening properties of Co/Cu(001) in its full realistic complexity.We solve an AIM for the whole Co 3d shell and we focus on the role of the parametrization of the Coulomb tensor for the Kondo effect.It is important to compare our findings to previous studies in the literature, in order to highlight both the differences and the similarities.
Previous theoretical analyses were restricted to a twoorbital model for the Kondo-active orbitals 15 , with approximate interaction schemes 13,15 or impurity solvers 13 .The most direct comparison can be done with the results reported by Jacob 13 , obtained with similar interaction parameters as ours, derived from first-principles within the constrained random-phase approximation.There, many-body effects are taken into account at the level of the one-crossing approximation (OCA), in contrast to our numerically exact CT-QMC.The OCA calculation takes into account all density-density terms as well as the spin-flip contributions.It may therefore be regarded as an intermediate parametrization between density-density and Kanamori, albeit restricted to one-crossing diagrams.There, a Kondo feature for the d z 2 orbital at T ∼ 10 K for Co/Cu(001), is reported, with a Kondo temperature T K ≈ 90 K, estimated from the width of the Kondo resonance in the spectral function.The lack of a similar feature for the other Kondo-active orbital (d xy in the notation of this work) was suggested as evidence of an underscreened Kondo effect.Whether the Co magnetic moment is completely screened at lower temperature, with the onset of a Fermi liquid state and the realization of a two-stage Kondo effect, was not investigated, and it remains debatable.
On the basis of our CT-QMC results we can delineate a quite different situation, whose physical explanation can be unveiled thanks to our comparative analysis of the various Coulomb tensor parametrizations.Within the density-density approximation the overall T K is much smaller than the lowest temperature of our calculation, and the Nevidomskyy-Coleman scenario for a spin S = 1 Kondo is fully realized.We progressively include additional exchange interactions within the Co 3d multiplet in the Kanamori and eventually all of them in the full Coulomb parametrizations.Due to the associated charge redistribution, spin fluctuations are partially quenched, whereas charge fluctuations increase, together with the orbital entanglement.Two effects consequently emerge.The d xy and d z 2 orbitals start to thrive on Kondo screening, especially with the full Coulomb interaction, while the remaining three d-orbitals substantially increase their active contribution to the local moment.The latter is transparently observed by comparing the charge distribution within the Co 3d multiplet in Fig. 5 and the contribution of the (d xy , d z 2 ) subspace to the local moment in Fig. 4 (by moving from the left to the right panels).The relevant role of the whole Co 3d shell within the full Coulomb parametrization of the interaction has also been suggested in the past 14,22 .However, the temperature regime previously investigated is hardly relevant for extracting useful information about the Kondo screening.
The outcome of the present study therefore changes the conventional interpretation of the Kondo effect in the prototypical Co-adatom systems, once a realistic interaction tensor is properly taken into account.Since the whole 3d shell is involved in the Kondo screening, one neither has a Nevidomskyy-Coleman scenario with the screening of a S = 1 spin at low temperatures, nor two independent S = 1/2 spin Kondo replicas in the (d xy , d z 2 ) subspace.The most appropriate way of describing the Kondo effect in Co adatoms on a Cu(001) surface is, as a matter of fact, a multi-orbital entangled correlated state.While two of the five 3d orbitals have the largest magnetic moment and a favorable hybridization to the substrate in order to display clear Kondo peaks, they are not decoupled enough from the other orbitals to allow for an effective two-orbital description of the Co 3d shell.
Finally, we note that some details of the calculations may differ from other results in the literature.For instance, Jacob 13 and Baruselli et al. 15 consider STM geometries, where the STM tip also consists of a Cu pyramid grown in the (001) direction -or the (111) direction when considering Co/Cu(111).In some cases 8,13,45 , besides the Co-Cu adsorption distance, also the atomic positions of some Cu atoms of the surface layer are relaxed.Despite these effects possibly being important, we are confident that the differences observed within the different parametrization of the Coulomb interaction influence the Kondo screening in a more fundamental way than the details of the DFT calculations.
To conclude, we revisited the prototypical Co/Cu(001) Kondo problem under a new light.We established how the parametrization of the Coulomb tensor affects the screening of the impurity magnetic moment, and we highlight the active role of the whole Co 3d shell in the Kondo effect.Our analysis is likely and can be extended to other Kondo systems with transition metal adatoms.
Throughout our study, we realized that the physical picture of Co/Cu(001) delicately depends on the size of the k-mesh of the Brillouin zone.In Fig. 8 we show the orbital-resolved (and spin-independent) hybridization function ∆(ıω n ) ≡ ∆ iσ (ıω n ) describing the embedding of the Co adatom on the Cu surface.We compare the results obtained for different k-meshes (all centered around the Γ point).We find that ∆(ıω n ) displays a slow convergence with the size of the k-mesh, in particular for the d z 2 orbital.For the sparsest mesh considered, i.e., 4 × 4 × 1 k-points, ∆(ıω n ) at low frequencies displays a qualitatively different behavior for the d z 2 orbital when compared to more accurate meshes (with up to 100 × 100 × 1 k-points), while for the d xy orbital the differences are mainly quantitative.Since the differences are observed at relatively low energy scales, it is possible that this effect may be overlooked in calculations with a low energy resolution, or with a large smearing parameter η in the hybridization function ∆(ω + ıη).Differences between the 40×40×1 and 100×100×1 k-points meshes can be observed on energy scales 0.01 eV, which corresponds approximatively to the lowest Matsubara frequency for the lowest temperatures of our QMC calulations, β = 350 − 1 (shaded area in Fig. 8).
However, here we show that this seemingly technical detail can have drastic consequences on the physical description.For instance, we can consider the temperature evolution of the lowest Matsubara frequency ω 0 of the electronic self-energy, which in a Fermi liquid should scale as Σ(ıω 0 ) ∝ T (see e.g., Refs. 40,46).In Fig. 9 we show Σ(ıω 0 ) for both the d xy and d z 2 orbitals within the density-density approximation.The Fermi liquid scaling seems to be recovered at low-enough temperatures for the sparsest 4 × 4 × 1 k-mesh, for both orbitals.For denser k-meshes, and in particular for the 100 × 100 × 1 one, the self-energy of the d z 2 orbital displays a clear non-Fermi liquid behavior, which we follow down to T ≈ 33 K.
Finally, in Fig. 10 we compare Σ(ıω 0 ) obtained within all parametrizations of the Coulomb tensor.We note that only the full Coulomb case seems to be compatible with a linear behavior, although this feature alone is not enough to confirm the onset of a Fermi liquid state at low temperatures.This observation is important because in the literature, calculations for Co/Cu(001) performed without including the full Coulomb tensors 13,15 indicated the Kondo screening to be the most effective for the d z 2 orbital.Our calculations show that, with an accurateenough description of the hybridization between the Co adatom and the Cu surface, and at low-enough temperatures, the density-density approximation does not confirm this picture.It is instead necessary to take into account more realistic forms of the Coulomb interaction to obtain estimates of the Kondo scale comparable with the experimental observations.

FIG. 1 .
FIG. 1. Top and side view of the unit cell of Co/Cu(001) as used in this study.

FIG. 2 .
FIG. 2. Possible Kondo scenarios for Co impurities on Cu hosts.In all three panels (a-c), the complexity of the Coulomb tensor increases from top to bottom.(a) Atomic ground state configuration of the Co 3d shell in the Cu(001) crystal field: (dxy, d z 2 ) half-filled orbitals identify the Kondo-active subspace, but for realistic Coulomb tensors the whole multiplet is relevant to the Kondo screening.(b) Schematic behavior of inter-orbital (i, j) spin and charge fluctuations in relation to the complexity of the Coulomb tensor.(c) Kondo screening processes (top to bottom).If the Co impurity is locked in a high-spin state (due to the Hund's coupling JH = Uijji) the Nevidomskyy-Coleman scenario18 can be realized at very low TK .As inter-orbital spin correlations are weaker and charge fluctuations are enhanced, an underscreened (or possibly two-stage) Kondo effect may take place at substantially higher TK .A collection of S = 1/2 replicas would be favored at JH = 0, but could also emerge due to the competition between all possible generalized exchange interactions (e.g., of the form U ijjk ) within the Co 3d shell.More complex SU(N > 2) spin-orbital Kondo scenarios, depending on the degree of orbital degeneracy, would also lead to an enhancement of the Kondo scale.

FIG. 3 .
FIG. 3. Orbital-resolved static spin-spin response function χ(T ) ≡ χii(T ) for the dxy and the d z 2 orbitals.The dashed lines shows the Curie-Weiss behavior χ(T ) ∝ 1/T in the local moment regime.[Inset] Plotting T χ(T ) highlights the differences observed with the three interaction schemes, with T χ(T ) ∼ const. in the local moment regime, and linearly vanishing at T TK (see text).

FIG. 4 .
FIG. 4. Analysis of the spin correlations at the Co impurity.[Upper panels] Unscreened (open symbols) and screened (filled symbols) effective spin S eff (τ ) estimated from the spin susceptibility (see text for the details).[Lower panels] Screened magnetic moment m 2 (β/2) within the whole 3d shell (filled symbols) and restricted to the (dxy, d z 2 ) subspace (solid line).The dashed lines separates intra-and inter-orbital contributions within the subspace.Due to the increasing contribution of orbitals outside the subspace, a two-Kondo-active orbital description of the magnetic moment becomes unsatisfactory for realistic Coulomb interactions.

FIG. 5 .
FIG. 5. Generalized double occupations niσn jσ for parallel (σ = σ) and anti-parallel (σ = σ) spin orientation for the Co 3d multiplet.The red borders highlight the rows and columns of the Kondo-active dxy and d z 2 orbitals.Data for T = 232 K, but within each interaction scheme the charge distribution is nearly independent of temperature down to T ≈ 30 K.

FIG. 6 .
FIG. 6. [Upper panels] Generalized double occupations within the density-density approximation.The 2 × 2 block represents the (dxy, d z 2 ) subspace while the extra block, labelled "rest", denotes the average over the diagonal elements for the (dyz, dxz, d x 2 −y 2 ) subspace.The left-hand (right-hand) hand side correspond to n d = 8 (n d = 8.2) electrons in the Co 3d shell.Moving away from integer filling results in a net charge transfer (in addition to the extra ∆n d = 0.2 electrons) to the (dxy, d z 2 ) subspace from the rest of the multiplet.The overall effect is a weakening of the tendency towards the S = 1 highspin state.[Lower panel] Spectral function A(ω) of the dxy and d z 2 orbitals within the density-density approximation.The development of low-energy resonances away from integer filling is compatible with an enhancement of the Kondo scale.

1 ,FIG. 7 .
FIG. 7. [Main panel] Orbital-resolved spectral function of the Co 3d shell at integer filling, i.e., n d = 8 electrons in the full Coulomb parametrization.The resonances of the (almost degenerate) dyz and dxz orbitals suggest more than two orbitals to be relevant to the Kondo effect.[Side panels] Temperature evolution of the low-energy resonance for the dxy and d z 2 orbitals.