Long range and highly tunable interaction between local spins coupled to a superconducting condensate

Interfacing magnetism with superconducting condensates is rapidly emerging as a viable route for the development of innovative quantum technologies. In this context, the development of rational design strategies to controllably tune the interaction between magnetic moments is crucial. Here we address this problem demonstrating the possibility of tuning the interaction between local spins coupled through a superconducting condensate with atomic scale precision. By using Cr atoms coupled to superconducting Nb, we use atomic manipulation techniques to precisely control the relative distance between local spins along distinct crystallographic directions while simultaneously sensing their coupling by scanning tunneling spectroscopy. Our results reveal the existence of highly anisotropic interactions, lasting up to very long distances, demonstrating the possibility of crossing a quantum phase transition by acting on the direction and interatomic distance between spins. The high tunability provides novel opportunities for the realization of topological superconductivity and the rational design of magneto-superconducting interfaces.

With the metallic tip, we observe a superconducting gap (∆ Nb = 1.53meV), which becomes doubled with the superconducting Nb tip, featuring very sharp coherence peaks and a sub-meV energy resolution. On a single isolated Cr atom with a metallic tip, a YSR resonance at zero bias is observed, which shows a higher spectral weight on the negative side only when a superconducting tip is used that shifts all spectral features about ±∆ tip . Temperature during measurement: 0.55K.  Figure  S2) can be clearly revealed. Although at the limit of our experimental energy resolution, the formation of bonding and anti-bonding YSR states of different orbital origin can be also revealed in most of the cases.  Fig. 4: Numerical simulation of the correlation between a small shift in the YSR energy ∆E YSR of ±100 meV around zero and the analysis parameter δ which was used in this work. Top panels: A dynes function combined with two peaks symmetric around, but close to, zero energy, convoluted with another dynes function was used to describe the dI/dU signal measured during the experiment with a superconducting tip. Middle panels: A small shift of YSR energy, below the limit of our energy resolution, results in a significant change of the in-gap peak intensities in the convoluted spectrum, enabling the detection of tiny coupling effects in the experiment. Bottom panels: The analysis parameters P + , P − (positive and negative peak intensity) and δ = P + − P − . In a small energy window around zero, there is approximately a linear correlation between δ and ∆E YSR .   The highly anisotropic behaviour can be traced back to the anisotropy of the Fermi surface. Good nesting vectors between flat parts of the Fermi surface can have a focusing effect on the wave function of quasi particles, significantly enhancing the coupling strength over larger distances. However, the discrete sampling induced by the lattice can lead to the aliasing effect between the periodicity of the lattice and the wavelength of the long-range oscillations, which complicates the identification of a specific region of the Fermi surface responsible for the oscillatory behavior. This is possible only in the asymptotic limit, i.e. at very large distances beyond those tracked experimentally, where the constructive interferences emanating from the nesting regions prevail. This is a well-known problem first highlighted within the context of indirectly coupled magnetic multilayers as discussed in Ref. [1].  The magnetic structure is obtained by minimizing the generalized Heisenberg model containing the magnetic on-site anisotropy K i of atom i, the isotropic exchange interaction J ij and the Dzyaloshinskii-Moriya interaction D ij . After performing ab-initio calculations, the pair interactions were calculated using the method of infinitesimal rotations [2,3]. In general, the magnetic coupling is weak, except for the nearest-neighbor dimers, which are not part of the experimental study. Therefore, we approximated the on-site anisotropy of each dimer atom by the one of an isolated Cr adatom, which was obtained using the method of constraining fields [4]. The effective angles α ij = arccos(e i · e j ) formed by the two dimer atoms are shown in Supplementary Figure 8a

Supplementary Note 9: Calculation of the YSR energies
Using the inverse of the Green function of the dimer obtained from first-principles, we construct the hybridization function of the dimer complexes which contains various information such as the hybridization strength with the surface, the crystal field splitting, the strength of spin-orbit coupling, and the strength of the direct hopping between the dimer atoms. The hamiltonian can be written as an effective tight-binding model, where the on-site part is given by (omitting the atom index i on the parameters) and the inter-atomic hopping part is given by E d is the average energy of the d-orbitals with respect to the Fermi energy, 2U represents the exchange splitting of the magnetic moment pointing along e, σ = (σ x , σ y , σ z ) is the vector of Pauli matrices, λ is the strength of the local spin-orbit coupling, L is the local orbital angular momentum operator, ∆ (re) is an orbital dependent energy shift corresponding to the crystal field splitting, Γ and ∆ (im) are non-hermitian contributions that result from the hybridization with the substrate, and t mm is the orbital-dependent hopping between atoms i and j.
Using the scheme described in the Supplementary Information of Ref. [5] the effective non-magnetic and magnetic scattering contributions, V m and J m , of the impurity-substrate s-d interaction I m can be obtained by virtue of the Schrieffer-Wolff transformation [6], which gives access to the energies of the YSR states [7,8], The parameters α m and β m can be directly obtained from ab-initio by diagonalizing the effective Hamiltonian construction given in eqs. (4) and (5), Note that this scheme is based on the eigenstates of a single atom. To account for the impact of the hopping t ij between the two dimer atoms, we repeat the effective Hamiltonian construction down to the single atom.
Using this procedure the local on-site parameters are renormalized by the hopping, which allows to quantify the impact of the hopping on the YSR energies. The bare parametrization obtained for the isolated Cr adatom (α d z 2 = 0.795 and β d z 2 = −0.140 ) can be found in the Supplement of Ref. [5] . The change of the YSR energies of the experimentally relevant orbital with z 2 character (m = 0) is shown in Supplementary Figure 9a   [001] Supplementary Fig. 9: Shift of the YSR energies of the z 2 -orbital of the Cr atoms based on various model assumption for all considered Cr dimers. The first three panels show the full tight-binding model with the realistic magnetic structure (a), as shown in Supplementary Figure 8, a ferromagnetic configuration for all dimers (b) and an antiferromagnetic configuration for all dimers (c). The impact of the coupling between the dimer atoms on the YSR energy is shown in the last two panels using the realistic magnetic structure. No direct hopping t ij is considered (d) and additionally also no renormalization of the orbital-dependent broadening ∆ (im) i is used, which can be seen as an indirect hopping effect (e). of the atomic relaxations or a renormalization due to zero-point quantum fluctuations [9], we used for that particular dimer in Figure 3 of the main manuscript the result obtained with the ferromagnetic coupling instead of the antiferromagnetic one. Supplementary Figure 9d and e show the impact of the direct hopping t ij and the orbital-dependent hybridization ∆ (im) using the realistic magnetic structure. For most dimers the direct hopping has a crucial contribution to the shift of the YSR energies. Only the dimer in the [001] direction shows still a significant shift without the direct hopping t ij , which vanishes if in addition ∆ (im) , which can be seen as an additional substrate-mediated effect of the coupling between the dimer atoms, is neglected (panel e). In conclusion, the main origin of the shifts in the YSR energies can be attributed to the nature of the magnetic coupling and the hopping between the dimer atoms.
To shed some more light on the former aspect, we use in the following the Alexander-Anderson model as described in Ref. [10], which is a two-site model with a single orbital per site and spin channel. The main idea is that the nature of the coupling influences the formation of bonding and anti-bonding states [11]. For a ferromagnetic coupling the majority (minority) spin channels of both atoms can hybridize leading to a double Lorentzian structure symmetrically shifted by ±t 12 , which in terms of our effective single-particle Hamiltonian can be viewed as an increased broadening Γ renorm = Γ + ∆Γ while the effective spin splitting In an effective single-particle single-orbital model (dashed line) it is reflected as an increased renormalized hybridization Γ renorm . Only one spin channel is shown for the sake of simplicity. b Antiferrormagnetic coupling. The coupling between the dimer atoms reflects itself as an effective renormalized spin splitting U renorm .
U remains unchanged (see illustration in Supplemenatry Figure 10a), which can be shown to be given by ∆Γ ∝ t 2 /Γ. In contrast, in the case of an anti-ferromagnetic coupling the majority spin channel of one atom couples to the minority spin channel of the other atom (and vice versa), which manifests itself in a shift of the energy levels and can be viewed as an increase of the effective spin splitting U renorm = U + ∆U and an unchanged broadening Γ in the single-particle description as illustrated in Supplementary Figure 10b. The change in the effective spin splitting is given by ∆U = t 2 /2U [10].
To analyze the impact of the magnetic coupling analytically, in the following we take the limit E m U and Γ U for which eq. (6) simplifies to with α m = −Γ/U . A change of Γ and U will modify α m in the following fashion which yields Since for the FM and AFM coupling ∆Γ > 0 and ∆U > 0, respectively, we expect from eqs. (9) and (10) that FM and AFM coupling induce shifts of the YSR energies in opposite directions, which agrees well with our findings using the full model shown in Supplementary Figure 9b