Multiple-scattering approach for multi-spin chiral magnetic interactions: Application to the one- and two-dimensional Rashba electron gas

Various multi-spin magnetic exchange interactions (MEI) of chiral nature have been recently unveiled. Owing to their potential impact on the realisation of twisted spin-textures, their implication in spintronics or quantum computing is very promising. Here, I address the long-range behavior of multi-spin MEI on the basis of a multiple-scattering formalism implementable in Green functions based methods. I consider the impact of spin-orbit coupling (SOC) as described in the one- (1D) and two-dimensional (2D) Rashba model, from which the analytical forms of the four- and six-spin interactions are extracted and compared to the bilinear isotropic, anisotropic and Dzyaloshinskii-Moriya interactions (DMI). Similarly to the DMI between two sites $i$ and $j$, there is a four-spin chiral vector perpendicular to the bond connecting the two sites. The oscillatory behavior of the MEI and their decay as function of interatomic distances are analysed and quantified for the Rashba surfaces states characterizing Au surfaces. The interplay of beating effects and strength of SOC gives rise to a wide parameter space where chiral MEI are more prominent than the isotropic ones. The multi-spin interactions for a plaquette of $N$ magnetic moments decay like $\{q_F^{N-d} P^{\frac{1}{2}(d-1)}L\}^{-1}$ simplifying to $\{q_F^{N-d} R^{\left[1+\frac{N}{2}(d-1)\right]}N\}^{-1}$ for equidistant atoms, where $d$ is the dimension of the mediating electrons, $q_F$ the Fermi wave vector, $L$ the perimeter of the plaquette while $P$ is the product of interatomic distances. This recovers the behavior of the bilinear MEI, $\{q_F^{2-d} R^{d}\}^{-1}$, and shows that increasing the perimeter of the plaquette weakens the MEI. More important, the power-law pertaining to the distance-dependent 1D MEI is insensitive to the number of atoms in the plaquette in contrast to the linear dependence associated with the 2D MEI.

Various multi-spin magnetic exchange interactions (MEI) of chiral nature have been recently unveiled. Owing to their potential impact on the realisation of twisted spin-textures, their future implication in spintronics or quantum computing is very promising. Here, I address the longrange behavior of multi-spin MEI on the basis of a multiple-scattering formalism implementable in Green functions based methods such as the Korringa-Kohn-Rostoker Green function framework. I consider the impact of spin-orbit coupling (SOC) as described in the one-(1D) and two-dimensional (2D) Rashba model, from which the analytical forms of the four-and six-spin interactions are extracted and compared to the well known bilinear isotropic, anisotropic and Dzyaloshinskii-Moriya interactions (DMI). Similarly to the DMI between two sites i and j, there is a four-spin chiral vector perpendicular to the bond connecting the two sites. The oscillatory behavior of the MEI and their decay as function of interatomic distances are analysed and quantified for the Rashba surfaces states characterizing Au surfaces. The interplay of beating effects and strength of SOC gives rise to a wide parameter space where chiral MEI are more prominent than the isotropic ones. The multi-spin interactions for a plaquette of N magnetic moments decay like {q N −d In a multiple-scattering formalism as the one utilized currently in the Korringa-Kohn-Rostoker (KKR) Greenfunction method, the Green function G describes the propagation of electron states from a given site i to a site j. The electron scattering at the atomic potentials V is quantified in terms of the single-ste scattering matrix t. Assuming the infinitesimal rotation of the magnetic moments allows to extract the MEI from a mapping procedure between the energy obtained from electronic structure calculations and the energy of an extended Heisenberg model [89][90][91] . The magnetic force theorem 92,93 permits to use the band energy instead of the total energy to evaluate the impact of rotating magnetic moments. This leads to a power series expansion: The trace Tr is taken over the sites i, orbital momentum L = (l, m), and spin indices while G and δt are matrices in angular momentum and spin representation, L ⊕ s. They respectively correspond to the structural Green function of the electronic system prior to the perturbation and the change of the single-site scattering matrix after rotating the magnetic moments. Note that one can work with the atomic potentials, δV , but then one uses the full Green functions instead of the structural ones.
Adopting the rigid spin approximation implies a change of the magnetic part of the single-site t-matrix, which can be expressed as Similarly to the proposal made in Ref. 91, t s i ( ) describes the scattering at the magnetic part of the atomic potential or of the exchange and correlation potential, B xc , if one works in the framework of density functional theory: where the integration is performed over the volume defining the atomic site i. R and R × are respectively the leftand right-hand wave functions resulting from the potential V . They are scattering solutions of either the Dirac or of the Schrödinger/scalar-relativistic equations augmented with the SOC. The Green function can be decomposed into two Green functions, A that is non-magnetic and diagonal in spin space and B that contains the magnetic part and a contribution induced by the spin-orbit coupling (SOC): where σ is the vector of Pauli matrices and σ 0 is the identity matrix. If inversion symmetry is not broken, A ij is symmetric with respect to site exchange and likewise for B σ ij if SOC is not present. B σ ij could have a magnetic contribution, which should behave like A ij , and a contribution from SOC. Within the Rashba model, where for example broken-inversion symmetry and SOC are incorporated, B σ ij = −B σ ji . Note that the decomposition of the Green functions as proposed by Cardias et al. 94 is useful for the investigation of multi-spin excitations.

A. Two-spin interaction terms
Considering infinitesimal rotations, we proceed to a Taylor expansion of the logarithm in Eq. (1) and extract systematically high-order terms ordered according to powers of δt. As aforementioned terms with odd powers of δt cancel since they are not compatible with the time-reversal symmetry requirement of the total energy. The standard second order term 4 leads to the well-known interaction energy between two magnetic moments from which the bilinear MEI can be extracted: The integrands are energy-dependent products of matrices of the same size than the inter-site Green functions and are given by the following forms: where the decomposition of the Green functions introduced in Eq. (4) leads to forms different from the usually presented ones 90,91 . Interestingly, this decomposition permits to identify two contributions to J: one is independent of magnetism and SOC while the other one depends on both with at least a quadratic dependence on SOC but being rotationally invariant in spin space owing to the inner product of the different B. D is, however, linear in B and consequently in SOC. The direction of the DMI vector is dictated by B and for lattices with inversion symmetry, we naturally recover that the DMI vanishes because of the symmetry of A and B with respect to site exchange.

B. Four-spin interaction terms
The fourth-order term in the Taylor expansion of the energy, involves plaquettes of four magnetic moments. We note that, in general, the four-spin contracted-site interaction terms can be recovered from the aforementioned forms by equaling some of the indices, for example by replacing the fourth site l with j to get the three-site terms, while replacing both (k, l) with (i, j) leads to the biquadratic terms. Note that this can lead to vanishing of some of the derived terms and the recovery of bilinear-like terms. Isotropic interaction. Analogous to the two-spin interaction, E 4-spin gives rise to the conventional isotropic fourspin interactions proportional to (S i · S j )(S k · S l ) including quadratic and fourth order contributions of the SOC. The isotropic change of the energy reads with which one recovers the usual form of the isotropic four-spin energy with in zero-order with respect to B. Four-spin chiral interactions. The fourth-order term in the Taylor expansion of the energy gives also rise to terms linear in the SOC. The latter are the recently derived 37 and postulated 41 four-spin vector-chiral interactions proportional to (S i · S j )(S k × S l ) and to the scalar spin-chirality S i · (S j × S k ) 42 . Besides these two terms, one additional term proportional to the three-spin vector chirality S i × (S j × S k ) shows up. We note that the different terms can be rewritten as function of each others. The product of the four Green-function elements in Eq. (13) and the requested linearity in B leads finally to four terms of the type BAAA, where B can be placed at four different positions relative to A. For instance, BAAA can be written in various ways depending on how the products involving the Pauli matrices are grouped: or where the bars on the indices indicate the sites connected by B. It is interesting to note that the integrand showing up in the previous equations can be interpreted as products involving the bilinear terms J iso and D (see Eqs. [10][11][12]. In Eq. 19, one can identify the three-spin scalar chirality, χ ijk = S i · (S j × S k ), and if I introduce what I call a three-spin vector chirality ℵ ijk = S i × (S j × S k ), the four-spin chiral energy linear in B reads This equation indicates that the ground state favored by such a four-spin chiral energy is the result of a complex competition of various terms: the three-spin scalar chirality, the three-spin vector chirality and a mix of cross and dot products. For instance, the three-spin scalar chirality pertaining to an equilateral triangular plaquette reaches its largest value for a polar angle cos(θ) = 1/ √ 3 while the largest length of the three-spin vector chirality is found for for cos(θ) = √ 7/3. By focusing on the first line of Eq. 20, one notices that the inner product Cīj kl · S j favors a parallel alignement between the four-spin chiral interaction vector and the magnetic moment j. The cross products S i × S j , however, drives the spin-configuration towards a state where the two perpendicular magnetic moments S j and S i live in a plane perpendicular to Cīj kl . The third term favors the chiral vector and the three magnetic moments at sites j, k, l to lie in the same plane perpendicular to S i with S k and S l perpendicular to each other while Cīj kl and S i would be parallel. Although, I present the four-spin energy as a sum of three types of terms, as shown in Ref. 43 one can rewrite all of them as function of the latter one.
While E B contains the first-order contributions in terms of B to E 4-spin , second-order contributions give rise to terms correcting the dot-product and cross-product of the magnetic moments: As it can be noticed in the following example, where I provide the contribution arising from BBAA, one finds products of bilinear terms similar to J ani multiplying J iso or D as defined in Eqs.10-12 : The first r.h.s. term looks like the anisotropic bilinear term (Eqs.12) weighted by a dot product between two magnetic moments of the four-spin plaquette. The second and the third terms can be rewritten in a form similar to the first one: and while the fourth term can be written as 6 C. Six-spin interaction term The sixth order term obtained from Eq. (1) involves a plaquette of six magnetic moments. We expect the largest contribution to be the isotropic, rotationally invariant one, as this term can be finite without spin-orbit interaction. Although being isotropic, one finds contributions involving products of three-spin scalar chiralities, or of three-spin vector chiralities: and thus with χ ijk = S i · (S j × S k ) and ℵ ijk = S i × (S j × S k ). In general, the effective 6-spin interaction is given by

III. RASHBA MODEL
Here I discuss the derived multi-spin interactions considering localized spins embedded in 1D and 2D electron gas, with spin-orbit coupling added in the form of Rashba coupling 78,79 . The conduction electrons are then either confined along a given direction (1D) or in the xy-plane (2D). The effective electric field emerging from spin-orbit coupling points along the z-axis. The Hamiltonian of both systems consists of the kinetic energy of the free electrons to which a linear term in momentum p is added: where α is the so-called Rashba parameter representing the strength of the spin-orbit interaction,ẑ is the unit vector along the z-axis, m * is the effective mass of the electron, σ 0 is the identity matrix and σ is the vector of Paul spin matrices.
Similarly to Ref. 70, the potential of each of the localized spins is described via the s − d interaction J sd , which has units of energy multiplying length to the power of dimensionality, such that the change of the magnetic part of the potential upon rotation of the spin moment is given by In the following, I use the s − d potential instead of the energy-dependent t-matrix in order to get analytical forms of the multi-spin interactions similarly to what is done in the RKKY approximation.

A. Rashba model -One dimension
The Rashba Green corresponding to the previous Hamiltonian for the 1DEG reads where r ij = r ij (cos β ij , sin β ij ) is the vector connecting sites i and j, whileβ ij = (sin β ij , − cos β ij ) is the unit vector perpendicular to r ij . Defining q = 2m 2 ε + k 2 R with k R = m * 2 α, the components of the Green functions are 70 : Now, I can proceed with the multi-spin interactions and start with the basic bilinear types 70,72 .

Two-spin interactions
Within the 1DEG Rashba model, we recover the bilinear magnetic interactions that were already derived in Ref. 70: and Within the RKKY approximation, the direction of the DMI is alongβ ij , i.e. perpendicular to the bond connecting the two sites i and j. Naturally, without SOC, D and J ani vanish. For small k R r ij , the DMI is linear with the Rashba parameter α while the compass-term interaction shows a quadratic dependence similarly to the SOC correction to the isotropic MEI. These dependencies change at large k R r ij , where the various bilinear interactions can be of the same order of magnitude with a decay dictated by the range function F , which is in fact the isotropic bilinear MEI before application of the SOC 95,96 : where Si() is the sine integral function. Owing to the asymptotic behavior of the sine function: .. , the range function behaves like: Thus, the bilinear interactions are characterized by various decays, r −1 ij being the lowest one, but interference effects are expected with the term proportional to r −2 ij . The wave length of the oscillations is expected to be complex depending on the interatomic distances. It is given by 2λ F = π/q F at small q F r before interference effects kick in, which originate from either or both the 1D nature of the electron gas (Eq. 37) and from SOC (Eqs. 33, 34 and 35).

Four-spin interactions
The isotropic four-spin interaction. As derived in Eq. 16, this interaction involves a product of four cosine functions that depend on k R : For small k R r, the isotropic four-spin interaction experiences corrections with even power of the Rashba coupling parameter α. The lowest dependence being quadratic in this regime, this implies a SOC contribution to the magnetic energy that is similar to the isotropic bilinear contribution . Also, and in analogy to the latter term, a range function takes care of the distance-dependent behavior of the four-spin interaction: Here I define L ijkl = r ij + r jk + r kl + r li as the perimeter of the 1D plaquette (ijkl). Using once more the asymptotic form of the Sine integral function, the long-range behavior of F (1D) ijkl simplifies to: 8 which indicates that the decay of the interaction is similar to the bilinear ones and is related to the distance spanned by the scattered electrons involved in the processes giving rise to the MEI. Depending on q F and on the magnitude of the s − d interaction, the isotropic four-spin interaction can be of the same order of magnitude than the bilinear MEI. Compared to the latter, the wavelength of the oscillations decreases for the four-spin interaction. Assuming magnetic moments equidistant by R, the range function becomes: The four-spin chiral interaction. The chiral interaction that is linear in B has the following form whose direction is perpendicular to the bond connecting the sites mediating the Green function B similarly to the DMI vector. Note that this is particular to the RKKY approximation based on the Rashba model since in general, the direction of Cīj kl can be more complex as demonstrated in Ref. 43. In comparison to the isotropic four-spin interaction, one of the cosines is replaced by a sine function. For small k R r, similarly to the DMI a linear dependence with respect to the Rashba coupling parameter α is expected for relatively short distances. Therefore, increasing SOC would permit to increase the amplitude of such chiral four-spin interaction. Since it is quadratic in B, the anisotropic chiral four-spin interaction involves two sine functions depending on α and therefore the short-distance behavior is quadratic with respect to SOC: This parametrizes the energy contributions such as the one given in Eq. 22, which in the Rashba model reads:

Six-spin interactions
The isotropic six-spin interaction is given by where which in the asymptotic long-range regime simplifies to For completeness, I give the form of the range function for equidistant atoms: Interestingly, the decay function is similar to that of the bilinear and four-spin interactions highlighting their potential relevance at large distances.
To summarize the 1D case, the multi-spin interactions similarly decay like L −1 . Interestingly the modification of q F can considerably affect the amplitude of the interactions owing to their q N −1 F dependence, N being the number of interacting magnetic moments defining the investigated plaquette. Moreover, the chiral multi-spin interactions are expected to be at short distances linear, at least, with the Rashba SOC.

B. Rashba model -Two dimension
In the 2D case, A and B defining the Green function are given by linear combinations of Hankel functions of zero and first order, respectively: Similarly to what was done for the 1D case, I proceed to the dervation of the multi-spin MEI in the RKKY approximation. The functional dependence with respect to the magnetic moments is not affected and all the physics is primarily encoded within the range functions, F (2D) ijkl , i.e. the MEI without the Rashba SOC, which are listed below in the asymptotic regimes, i.e. when qr 1 and k R q. There, the Hankel functions are and Two-spin interactions. Here the two-spin range function gives the usual decay function proportional to r 2 ij for the 2D electron gas. Four-spin interactions. In the asymptotic regime the four-spin range function is similar to the one characterizing the two-spin 1D interactions with the already defined perimeter L ijkl = r ij + r jk + r kl + r li and the product of interatomic distances P ijkl = r ij r jk r kl r li . The slowest decay function of the four-spin interactions is given by ( √ P L) −1 , which simiplifies to R −3 when assuming a plaquette of equidistant atoms (r ij = r jk = r kl = r li = R ), in which case the range function becomes Thus and in contrast to the 1D-case, the four-spin interactions are expected to decay faster than the bilinear ones. A similar observation can be made for the following six-spin interactions.
Six-spin interactions. Here, the range function is approximately given by and if the atoms in the plaquette are equidistant as they would be in an hexagonal lattice (r ij = r jk = r kl = r lm = r mn = r ni = R ), the range function becomes One notices that decay function pertaining to the 2D interactions is given by (P 1 2 L) −1 which is different from the 1D case because of the additional factor P 1 2 that simplifies to R N 2 for N equidistant magnetic moments. This ignites a strong difference in the long-range behavior since the power-law decay associated to the 1D electon gas is independent from the number of atomic spins, while it increases linearly in the 2D case. Similarly to the 1D interactions, the higher-order 2D magnetic interactions are expected to change when modifying q F but the power-law dependence is . This points to the possibility of manipulating the impact and magnitude of the magnetic interactions by controlling either the position of the magnetic moments or the Fermi energy.
To summarize, one can state that in general the decay of the multi-spin MEI can be cast into the following formula with which the behavior of the two-spin interactions is recovered (≈ q d−2 F R −d ): with d = 1 or 2 depending on the dimension of the electron gas mediating the interactions. Noteworthy is the impact of SOC on amplitude of the chiral MEI. It is expected to be at least linear with α independently from the nature of the electron gas. It is important to keep in mind, that the contribution of the multi-spin interactions to the total energy increases in a factorial fashion when increasing the number of magnetic moments of the plaquettes. Therefore, although the coefficients defining the MEI might be smaller than the bilinear ones, their overall impact on the energy can be prominent. The parameter J sd is chosen such to give the right order of magnitude of the first minimum of the bilinear RKKY interaction, which corresponds to an antiferromagnetic coupling of two magnetic moments. To allow a simple comparison between the 1D and 2D cases, the first largest antiferromagnetic coupling is assumed to be the same and equal to 1 meV, which is about the value measured for the RKKY interactions between Co adatoms on Pt(111) surface 85 . I start by analysing the isotropic magnetic interactions without the Rashba SOC shown in Figs. 1(a-b). Note that for the bilinear interactions, a positive (negative) value corresponds to a ferromagnetic (antiferromagnetic) coupling. As expected, the number of oscillations increases with the number of sites involved in the interactions since the wavelength is given by 2 π N q F = λ F N , N being the number of spins involved in the interaction while R is the interatomic distance, which for simplicity is assumed to be identical for all pairs of magnetic moments. This gives rise to situations where the higher order MEI are larger in magnitude than the bilinear ones. As expected, the 2D MEI decay more strongly than the 1D ones. This advocates for the investigation of the long-range multi-spin interactions in systems with reduced dimensionality.
As illustrated in Figs. 1(c-d) (see also Fig. 2(d-f) for a better resolution of the six-spin interaction), the inclusion of SOC reduces the amplitude of the isotropic multi-spin interactions with the emergence of a beating effect leading to a vanishing of the interactions accompanied with a phase switch of their oscillations, which is well known for the two-spin interactions 72 . This occurs at a length scale defined by the Rashba wave vector, k R , i.e. around 6 nm.
In Figs. 2(a-d), the bilinear MEI are shown for the spins embedded respectively in the 1D and 2D Rashba electron gas. Similarly to the isotropic MEI, the amplitude of DMI and J ani decreases by increasing the dimension of the electron gas. Making the moments further apart eventually favors the enhancement of the SO-driven interactions, which contrast strongly with the evanescence of the isotropic interactions around 6 nm.
The four-spin interactions are displayed in Figs. 2(b-e). These are initially, i.e. for relatively short interatomic distances, the largest in 2D but they experience a faster weakening than those of the 1D Rashba gas. Note that in contrast to the bilinear counterparts, the chiral four-spin interactions oscillate in opposite phase with respect to that of the isotropic ones.
It is instructive to compare the behavior of the chiral bilinear with that of the equivalent four-spin interactions (see Fig. 3). Although the amplitude of the four-spin interactions is weaker than that of the bilinear MEI, one notices that at short distances, their sign is opposite and owing to their different wavelengths, there are regions where the chirality behavior is settled by the four-spin interaction. For instance, this occurs at ≈ 1nm in the 2D case, where the isotropic MEI is negligible (see Figs. 1(b-d).
is chosen such to give the right order of magnitude of the first minimum of the bilinear RKKY interaction, which corresponds to an antiferromagnetic coupling of two magnetic moments. To allow a simple comparison between the 1D and 2D cases, the first antiferromagnetic coupling is assumed to be the same and equal to 1 meV, which is about the value measured for the RKKY interactions between Co adatoms on Pt(111) 58 . We start by analysing the isotropic magnetic interactions without the Rashba SOC shown in Fig.??. We notice that the number of oscillations increases with the number of sites involved in the interactions. This is expected from the argument of the oscillatory function which is given by Nq F R, N being the number of spins involved in the interaction while R is the interatomic distance, which for simplicity is assumed to be identical for all pairs of magnetic moments. This gives rise to situations where the higher order MEI are larger in magnitude than the bilinear ones. As expected, the MEI decay more strongly in the 2D than in the 1D case. This advocates for the investigation of the long-range multi-spin interactions in systems with a reduced dimensionality.
As illustrated in Fig.??, the inclusion of SOC reduces all of the isotropic multi-spin interactions with the emergence of a beating e↵ect leading to a vanishing of the interactions accompanied with a phase switch of their oscillations, which is well known for bilienar interactions 52 . This occurs at a length scale defined by the Rashba wave vector, k R , i.e. around 6 nm.
In Fig??, the bilinear MEI are shown for the spins embedded respectively in the 1D and 2D Rashba electron gas. Similarly to the isotropic MEI, the DMI and J ani in the 1D case are larger in magntiude than in the 2D. Increasing the interatomic distance favors the enhancement of the SO-driven interactions, which contrast strongly with the evanescence of the isotropic interactions around 6 nm.
The four-spin interactions are displayed in Fig.??. These are initially the largest in 2D case for relatively short interatomic distances but they experience a faster weakening than those of the 1D Rashba gas. In contrast to the bilinear counterparts, the chiral four-spin interactions oscillate in opposite phase with respect to that of the isotropic ones.
It is instructive to compare the behavior of the chiral bilinear with that of the equivalent 4-spin interactions. Although the magnitude of the 4-spin interactions is weaker together with a stronger decay than that of the bilinear MEI, one notices that a short distances, their sign is opposite and owing to their di↵erent wavelengths, there are regions where the chirality behavior is settled by the four-spin interaction, for instance at ⇡ 1nm in the 2D case.
is chosen such to give the right orde corresponds to an antiferromagnetic c 1D and 2D cases, the first antiferroma the value measured for the RKKY int We start by analysing the isotropic the number of oscillations increases w argument of the oscillatory function w while R is the interatomic distance, w This gives rise to situations where the the MEI decay more strongly in the multi-spin interactions in systems wit As illustrated in Fig.??, the inclusio of a beating e↵ect leading to a vanis which is well known for bilienar inter i.e. around 6 nm.
In Fig??, the bilinear MEI are show Similarly to the isotropic MEI, the D the interatomic distance favors the e evanescence of the isotropic interactio The four-spin interactions are disp interatomic distances but they exper bilinear counterparts, the chiral four-s ones.
It is instructive to compare the b Although the magnitude of the 4-spin MEI, one notices that a short distan regions where the chirality behavior i   The four-spin interactions are d interatomic distances but they exp bilinear counterparts, the chiral fou ones.
It is instructive to compare the Although the magnitude of the 4-s MEI, one notices that a short dis regions where the chirality behavio the interatomic distance favors the enhancement of the SO-driven interactions, which contrast strongly wi evanescence of the isotropic interactions around 6 nm.
The four-spin interactions are displayed in Fig.??. These are initially the largest in 2D case for relatively interatomic distances but they experience a faster weakening than those of the 1D Rashba gas. In contrast bilinear counterparts, the chiral four-spin interactions oscillate in opposite phase with respect to that of the iso ones.
It is instructive to compare the behavior of the chiral bilinear with that of the equivalent 4-spin intera Although the magnitude of the 4-spin interactions is weaker together with a stronger decay than that of the b MEI, one notices that a short distances, their sign is opposite and owing to their di↵erent wavelengths, the regions where the chirality behavior is settled by the four-spin interaction, for instance at ⇡ 1nm in the 2D ca We start by analysing the isotropic magnetic interactions without the Rashba SOC show the number of oscillations increases with the number of sites involved in the interactions. argument of the oscillatory function which is given by Nq F R, N being the number of spins while R is the interatomic distance, which for simplicity is assumed to be identical for all p This gives rise to situations where the higher order MEI are larger in magnitude than the the MEI decay more strongly in the 2D than in the 1D case. This advocates for the inve multi-spin interactions in systems with a reduced dimensionality.
As illustrated in Fig.??, the inclusion of SOC reduces all of the isotropic multi-spin inter of a beating e↵ect leading to a vanishing of the interactions accompanied with a phase s which is well known for bilienar interactions 52 . This occurs at a length scale defined by th i.e. around 6 nm.
In Fig??, the bilinear MEI are shown for the spins embedded respectively in the 1D an Similarly to the isotropic MEI, the DMI and J ani in the 1D case are larger in magntiude the interatomic distance favors the enhancement of the SO-driven interactions, which evanescence of the isotropic interactions around 6 nm.
The four-spin interactions are displayed in Fig.??. These are initially the largest in 2 interatomic distances but they experience a faster weakening than those of the 1D Rash bilinear counterparts, the chiral four-spin interactions oscillate in opposite phase with resp ones.
It is instructive to compare the behavior of the chiral bilinear with that of the equ Although the magnitude of the 4-spin interactions is weaker together with a stronger deca MEI, one notices that a short distances, their sign is opposite and owing to their di↵er regions where the chirality behavior is settled by the four-spin interaction, for instance at We start by analysing the the number of oscillations i argument of the oscillatory while R is the interatomic d This gives rise to situations the MEI decay more strong multi-spin interactions in s As illustrated in Fig.??, t of a beating e↵ect leading which is well known for bili i.e. around 6 nm.
In Fig??, the bilinear ME Similarly to the isotropic M the interatomic distance fa evanescence of the isotropic The four-spin interaction interatomic distances but t bilinear counterparts, the c ones.
It is instructive to comp Although the magnitude of MEI, one notices that a sh regions where the chirality         66 . The magnetic moments are assumed to be identical and equidistant to facilitate the discussion. The s − d interaction J sd is chosen such that for both the 1D and 2D electron gas, the bilinear magnetic exchange interaction at the first minimum of the oscillatory function equals -1 meV, favoring an antiferromagnetic coupling. This is motivated by measurements of the long-range magnetic interactions of Co adatoms on Pt(111) surface 85 .

IV. CONCLUSION
In this article, I present a theoretical framework for the evaluation of multi-spin interactions utilizing multiplescattering theory. This fits methodologies based on the calculation of Green functions, such as the Korringa-Kohn-Rostoker Green functions. I discuss the bilinear, four-and six-spin interactions with a particular focus on isotropic and chiral terms. Then I use this theory for the evaluation of multi-spin interactions for localized spins embedded in oneand two-dimensional electrons described by the Rashba model. Utilizing the RKKY approximation, the latter model offers the possibility of extracting analytically the long-range/asymptotic behavior of isotropic and chiral multi-spin interactions.
Within this approach, there is for each couple of sites i and j a four-spin chiral vector perpendicular to the bond connecting the two sites similarly to their DMI vector. The reported study shows that the strong contrast between the 1D and 2D bilinear magnetic exchange interactions survives for higher-order interactions. I recover the power-law decay pertaining to the two-spin magnetic exchange interactions, ≈ q d−2 F R −d , which we generalize to the N -spin case as {q where d is the dimension of the electron gas mediating the interactions, L the perimeter of the plaquette of N spins while P is the product of interatomic distances and q F the Fermi wave vector. The 2-, 4-and 6-spin 1D MEI experience a similar same decay, i.e. same power law, with respect

6-spin interactions (meV)
corresponds to an antiferromagnetic coupling of two magnetic moments. To allow a simple comparison between the 1D and 2D cases, the first antiferromagnetic coupling is assumed to be the same and equal to 1 meV, which is about the value measured for the RKKY interactions between Co adatoms on Pt(111) 58 .
We start by analysing the isotropic magnetic interactions without the Rashba SOC shown in Fig.??. We notice that the number of oscillations increases with the number of sites involved in the interactions. This is expected from the argument of the oscillatory function which is given by NqF R, N being the number of spins involved in the interaction while R is the interatomic distance, which for simplicity is assumed to be identical for all pairs of magnetic moments. This gives rise to situations where the higher order MEI are larger in magnitude than the bilinear ones. As expected, the MEI decay more strongly in the 2D than in the 1D case. This advocates for the investigation of the long-range multi-spin interactions in systems with a reduced dimensionality.
As illustrated in Fig.??, the inclusion of SOC reduces all of the isotropic multi-spin interactions with the emergence of a beating e↵ect leading to a vanishing of the interactions accompanied with a phase switch of their oscillations, which is well known for bilienar interactions 52 . This occurs at a length scale defined by the Rashba wave vector, kR, i.e. around 6 nm.
In Fig??, the bilinear MEI are shown for the spins embedded respectively in the 1D and 2D Rashba electron gas. Similarly to the isotropic MEI, the DMI and J ani in the 1D case are larger in magntiude than in the 2D. Increasing the interatomic distance favors the enhancement of the SO-driven interactions, which contrast strongly with the evanescence of the isotropic interactions around 6 nm.
The four-spin interactions are displayed in Fig.??. These are initially the largest in 2D case for relatively short interatomic distances but they experience a faster weakening than those of the 1D Rashba gas. In contrast to the bilinear counterparts, the chiral four-spin interactions oscillate in opposite phase with respect to that of the isotropic ones.
It is instructive to compare the behavior of the chiral bilinear with that of the equivalent 4-spin interactions. Although the magnitude of the 4-spin interactions is weaker together with a stronger decay than that of the bilinear MEI, one notices that a short distances, their sign is opposite and owing to their di↵erent wavelengths, there are regions where the chirality behavior is settled by the four-spin interaction, for instance at ⇡ 1nm in the 2D case.
while R is the interatomic distance, which for simplicity is assumed to be identical for all pairs of magnetic moments. This gives rise to situations where the higher order MEI are larger in magnitude than the bilinear ones. As expected, the MEI decay more strongly in the 2D than in the 1D case. This advocates for the investigation of the long-range multi-spin interactions in systems with a reduced dimensionality.
As illustrated in Fig.??, the inclusion of SOC reduces all of the isotropic multi-spin interactions with the emergence of a beating e↵ect leading to a vanishing of the interactions accompanied with a phase switch of their oscillations, which is well known for bilienar interactions 52 . This occurs at a length scale defined by the Rashba wave vector, kR, i.e. around 6 nm.
In Fig??, the bilinear MEI are shown for the spins embedded respectively in the 1D and 2D Rashba electron gas. Similarly to the isotropic MEI, the DMI and J ani in the 1D case are larger in magntiude than in the 2D. Increasing the interatomic distance favors the enhancement of the SO-driven interactions, which contrast strongly with the evanescence of the isotropic interactions around 6 nm.
The four-spin interactions are displayed in Fig.??. These are initially the largest in 2D case for relatively short interatomic distances but they experience a faster weakening than those of the 1D Rashba gas. In contrast to the bilinear counterparts, the chiral four-spin interactions oscillate in opposite phase with respect to that of the isotropic ones.
It is instructive to compare the behavior of the chiral bilinear with that of the equivalent 4-spin interactions. Although the magnitude of the 4-spin interactions is weaker together with a stronger decay than that of the bilinear MEI, one notices that a short distances, their sign is opposite and owing to their di↵erent wavelengths, there are regions where the chirality behavior is settled by the four-spin interaction, for instance at ⇡ 1nm in the 2D case.
corresponds to an antiferromagnetic coupling of two magnetic moments. To allow a simple comparison between the 1D and 2D cases, the first antiferromagnetic coupling is assumed to be the same and equal to 1 meV, which is about the value measured for the RKKY interactions between Co adatoms on Pt(111) 58 .
We start by analysing the isotropic magnetic interactions without the Rashba SOC shown in Fig.??. We notice that the number of oscillations increases with the number of sites involved in the interactions. This is expected from the argument of the oscillatory function which is given by NqF R, N being the number of spins involved in the interaction while R is the interatomic distance, which for simplicity is assumed to be identical for all pairs of magnetic moments. This gives rise to situations where the higher order MEI are larger in magnitude than the bilinear ones. As expected, the MEI decay more strongly in the 2D than in the 1D case. This advocates for the investigation of the long-range multi-spin interactions in systems with a reduced dimensionality.
As illustrated in Fig.??, the inclusion of SOC reduces all of the isotropic multi-spin interactions with the emergence of a beating e↵ect leading to a vanishing of the interactions accompanied with a phase switch of their oscillations, which is well known for bilienar interactions 52 . This occurs at a length scale defined by the Rashba wave vector, kR, i.e. around 6 nm.
In Fig??, the bilinear MEI are shown for the spins embedded respectively in the 1D and 2D Rashba electron gas. Similarly to the isotropic MEI, the DMI and J ani in the 1D case are larger in magntiude than in the 2D. Increasing the interatomic distance favors the enhancement of the SO-driven interactions, which contrast strongly with the evanescence of the isotropic interactions around 6 nm.
The four-spin interactions are displayed in Fig.??. These are initially the largest in 2D case for relatively short interatomic distances but they experience a faster weakening than those of the 1D Rashba gas. In contrast to the bilinear counterparts, the chiral four-spin interactions oscillate in opposite phase with respect to that of the isotropic ones.
It is instructive to compare the behavior of the chiral bilinear with that of the equivalent 4-spin interactions. Although the magnitude of the 4-spin interactions is weaker together with a stronger decay than that of the bilinear MEI, one notices that a short distances, their sign is opposite and owing to their di↵erent wavelengths, there are regions where the chirality behavior is settled by the four-spin interaction, for instance at ⇡ 1nm in the 2D case.
argument of the oscillatory function which is given by NqF R, N being the number of spins involved in the interaction while R is the interatomic distance, which for simplicity is assumed to be identical for all pairs of magnetic moments. This gives rise to situations where the higher order MEI are larger in magnitude than the bilinear ones. As expected, the MEI decay more strongly in the 2D than in the 1D case. This advocates for the investigation of the long-range multi-spin interactions in systems with a reduced dimensionality.
As illustrated in Fig.??, the inclusion of SOC reduces all of the isotropic multi-spin interactions with the emergence of a beating e↵ect leading to a vanishing of the interactions accompanied with a phase switch of their oscillations, which is well known for bilienar interactions 52 . This occurs at a length scale defined by the Rashba wave vector, kR, i.e. around 6 nm.
In Fig??, the bilinear MEI are shown for the spins embedded respectively in the 1D and 2D Rashba electron gas. Similarly to the isotropic MEI, the DMI and J ani in the 1D case are larger in magntiude than in the 2D. Increasing the interatomic distance favors the enhancement of the SO-driven interactions, which contrast strongly with the evanescence of the isotropic interactions around 6 nm.
The four-spin interactions are displayed in Fig.??. These are initially the largest in 2D case for relatively short interatomic distances but they experience a faster weakening than those of the 1D Rashba gas. In contrast to the bilinear counterparts, the chiral four-spin interactions oscillate in opposite phase with respect to that of the isotropic ones.
It is instructive to compare the behavior of the chiral bilinear with that of the equivalent 4-spin interactions. Although the magnitude of the 4-spin interactions is weaker together with a stronger decay than that of the bilinear MEI, one notices that a short distances, their sign is opposite and owing to their di↵erent wavelengths, there are regions where the chirality behavior is settled by the four-spin interaction, for instance at ⇡ 1nm in the 2D case.
interatomic distances but they experience a faster weakening than those of the 1D Rashba gas. In contrast to the bilinear counterparts, the chiral four-spin interactions oscillate in opposite phase with respect to that of the isotropic ones.
It is instructive to compare the behavior of the chiral bilinear with that of the equivalent 4-spin interactions. Although the magnitude of the 4-spin interactions is weaker together with a stronger decay than that of the bilinear MEI, one notices that a short distances, their sign is opposite and owing to their di↵erent wavelengths, there are regions where the chirality behavior is settled by the four-spin interaction, for instance at ⇡ 1nm in the 2D case.
MEI, one notices that a short distances, their sign is opposite and owing to their di↵erent wavelengths, there are regions where the chirality behavior is settled by the four-spin interaction, for instance at ⇡ 1nm in the 2D case.
bilinear counterparts, the chiral four-spin interactions oscillate in opposite phase with respect to that of the isotropic ones.
It is instructive to compare the behavior of the chiral bilinear with that of the equivalent 4-spin interactions. Although the magnitude of the 4-spin interactions is weaker together with a stronger decay than that of the bilinear MEI, one notices that a short distances, their sign is opposite and owing to their di↵erent wavelengths, there are regions where the chirality behavior is settled by the four-spin interaction, for instance at ⇡ 1nm in the 2D case.
multi-spin interactions in systems with a reduced dimensionality.
As illustrated in Fig.??, the inclusion of SOC reduces all of the isotropic multi-spin interactions with the emergence of a beating e↵ect leading to a vanishing of the interactions accompanied with a phase switch of their oscillations, which is well known for bilienar interactions 52 . This occurs at a length scale defined by the Rashba wave vector, kR, i.e. around 6 nm.
In Fig??, the bilinear MEI are shown for the spins embedded respectively in the 1D and 2D Rashba electron gas. Similarly to the isotropic MEI, the DMI and J ani in the 1D case are larger in magntiude than in the 2D. Increasing the interatomic distance favors the enhancement of the SO-driven interactions, which contrast strongly with the evanescence of the isotropic interactions around 6 nm.
The four-spin interactions are displayed in Fig.??. These are initially the largest in 2D case for relatively short interatomic distances but they experience a faster weakening than those of the 1D Rashba gas. In contrast to the bilinear counterparts, the chiral four-spin interactions oscillate in opposite phase with respect to that of the isotropic ones.
It is instructive to compare the behavior of the chiral bilinear with that of the equivalent 4-spin interactions. Although the magnitude of the 4-spin interactions is weaker together with a stronger decay than that of the bilinear MEI, one notices that a short distances, their sign is opposite and owing to their di↵erent wavelengths, there are regions where the chirality behavior is settled by the four-spin interaction, for instance at ⇡ 1nm in the 2D case.
bilinear counterparts, the chiral four-spin interactions oscillate in opposite phase with respect to that of the iso ones.
It is instructive to compare the behavior of the chiral bilinear with that of the equivalent 4-spin intera Although the magnitude of the 4-spin interactions is weaker together with a stronger decay than that of the b MEI, one notices that a short distances, their sign is opposite and owing to their di↵erent wavelengths, the regions where the chirality behavior is settled by the four-spin interaction, for instance at ⇡ 1nm in the 2D ca The four-spin interactions are displayed in Fig.??. These are initially the largest in 2D case for relatively short interatomic distances but they experience a faster weakening than those of the 1D Rashba gas. In contrast to the bilinear counterparts, the chiral four-spin interactions oscillate in opposite phase with respect to that of the isotropic ones.
It is instructive to compare the behavior of the chiral bilinear with that of the equivalent 4-spin interactions. Although the magnitude of the 4-spin interactions is weaker together with a stronger decay than that of the bilinear MEI, one notices that a short distances, their sign is opposite and owing to their di↵erent wavelengths, there are regions where the chirality behavior is settled by the four-spin interaction, for instance at ⇡ 1nm in the 2D case.
It is instructive to compare the behavior of the chiral bilinear with that of the equivalent 4-spin int Although the magnitude of the 4-spin interactions is weaker together with a stronger decay than that of th MEI, one notices that a short distances, their sign is opposite and owing to their di↵erent wavelengths, regions where the chirality behavior is settled by the four-spin interaction, for instance at ⇡ 1nm in the 2D        which is well known for bilienar interactions 5 i.e. around 6 nm.
In Fig??, the bilinear MEI are shown for t Similarly to the isotropic MEI, the DMI and the interatomic distance favors the enhance evanescence of the isotropic interactions arou The four-spin interactions are displayed i interatomic distances but they experience a bilinear counterparts, the chiral four-spin int ones.
It is instructive to compare the behavior Although the magnitude of the 4-spin intera MEI, one notices that a short distances, th regions where the chirality behavior is settle is chosen such to give the right order of m corresponds to an antiferromagnetic coupl 1D and 2D cases, the first antiferromagnet the value measured for the RKKY interact We start by analysing the isotropic magn the number of oscillations increases with t argument of the oscillatory function which while R is the interatomic distance, which This gives rise to situations where the high the MEI decay more strongly in the 2D th multi-spin interactions in systems with a r As illustrated in Fig.??, the inclusion of of a beating e↵ect leading to a vanishing which is well known for bilienar interaction i.e. around 6 nm.
In Fig??, the bilinear MEI are shown fo Similarly to the isotropic MEI, the DMI a the interatomic distance favors the enhan evanescence of the isotropic interactions ar The four-spin interactions are displayed interatomic distances but they experience bilinear counterparts, the chiral four-spin i ones.
It is instructive to compare the behavi Although the magnitude of the 4-spin inte MEI, one notices that a short distances, regions where the chirality behavior is sett     Similarly to the isotropic MEI, the DMI an the interatomic distance favors the enhan evanescence of the isotropic interactions aro The four-spin interactions are displayed interatomic distances but they experience bilinear counterparts, the chiral four-spin in ones.
It is instructive to compare the behavio Although the magnitude of the 4-spin inter MEI, one notices that a short distances, t regions where the chirality behavior is settl   to the spatial separation of the spins. In contrast, the more spins involved in the 2D MEI, the stronger is the decay. Moreover, the dependence with respect to q F provides a path of engineering the magnitude of the higher-order MEI by tuning the electronic occupation.
Regarding relativistic effects, obviously the chiral multi-spin interactions cancel out without SOC. The smallest dependence with respect to the latter is linear at short distances. At atomic separations of the order of SOC length scale, beating effects reduce the amplitude of the isotropic MEI giving the opportunity for the chiral interactions to have a strong impact on the magnetic behavior of the investigated systems.
Numerical results were presented on the basis of Rashba parameters mimicking surface states residing on Au surfaces. This permits a visual comparison of the various interactions. Their distinct oscillatory behavior offers the possibility of exploring rich magnetic phase diagrams, which can be strongly altered when modifying the interatomic distances. This is probably possible with atomic manipulation based on scanning tunneling microscopy. Additional tuning parameters are the strength of SOC and the electronic occupation, which can be changed by gating or by modifying the nature of the substrate 97-99 . In the future, it would be interesting to extend the current study to the 3D case, either by using a 3D Rashba model 100 or by extending the Fert-Levy model 69 to the multi-spin interactions. Decay is stronger in 2D than 1D. In 1D, Jani revives in the region where the beating effect occurs for Isotropic interaction. This is induced by the factor (1-cos) instead of cos for Jiso. The 4-spin chiral in are more relevant at relatively short distances. In that regime, they are dephased by about pi with r Ones meaning they would counter each other in terms of chirality. The dephasing for the 2D case is than pi, leading to regions where the 2-spin ones are negligible with respect to the 4-spin ones, i.e. MEI, one notices that a short distances, their sign is opposite and owing to their di↵erent wavelengths, there a regions where the chirality behavior is settled by the four-spin interaction, for instance at ⇡ 1nm in the 2D case.     For the 2D Rashba electrons, the dephasing is not perfect at relatively short distances, e.g. at 1 nm, which permits to have a chiral four-spin interaction that is much larger than the DMI. Note that the results were obtained with parameters of the surface state of Au(111) 66 .