The chiral biquadratic pair interaction

Magnetic interactions underpin a plethora of magnetic states of matter, hence playing a central role both in fundamental physics and for future spintronic and quantum computation devices. The Dzyaloshinskii-Moriya interaction, being chiral and driven by relativistic effects, leads to the stabilization of highly-noncollinear spin textures such as skyrmions, which thanks to their topological nature are promising building blocks for magnetic data storage and processing elements. Here, we reveal and study a new chiral pair interaction, which is the biquadratic equivalent of the Dzyaloshinskii-Moriya interaction. First, we derive this interaction and its guiding principles from a microscopic model. Second, we study its properties in the simplest prototypical systems, magnetic dimers deposited on various substrates, resorting to systematic first-principles calculations. Lastly, we discuss its importance and implications not only for magnetic dimers but also for extended systems, namely one-dimensional spin spirals and complex two-dimensional magnetic structures, such as a nanoskyrmion lattice.


Introduction
Starting from the seminal work of Heisenberg 1 , magnetic materials are often described by bilinear isotropic magnetic interactions, J ij S i · S j . However, a wealth of complex spin-textures were discovered over the last century that called for the enrichment of the original Heisenberg model with various other types of interactions (see e.g. Refs. [2][3][4][5][6][7][8][9][10][11] ). The magnetism of 3 He is a striking example, being dominated by higher-order isotropic interactions 12 which can be derived from the Hubbard model at half-filling [13][14][15] or from Kondo-lattice models [16][17][18] . These interactions, such as the biquadratic interaction B ij (S i · S j ) 2 and the related three-and four-site interactions, introduce nonlinear effects into the Heisenberg model. An important consequence is that different spin spirals, characterized by a wavevector Q, can be combined into lower-energy multiple-Q-states, as the higher-order interactions invalidate the superposition principle. Prominent examples are the antiferromagnetic uudd-state (a 2Q-state) 19,20 and the 3Q-state 21 . Interestingly, this 3Q-state (also magnetic skyrmions 22,23 and bobbers 24,25 ) is a noncoplanar magnetic state that hosts interesting Berry-phase physics arising from its non-vanishing scalar spin chirality S i · (S j × S k ), such as topological orbital ferromagnetism and Hall effects [26][27][28][29][30] .
The concept of vector spin chirality is embodied by the antisymmetric bilinear Dzyaloshinskii-Moriya interaction (DMI), D ij · (S i × S j ) 3, 4 , which arises due to the combination of spin-orbit coupling and absence of spatial inversion symmetry. The DMI lifts the energy degeneracy of magnetic spirals with opposite vector spin chirality, S i × S j , thus stabilizing magnetic structures of well-defined rotational sense, such as chiral spin spirals 31,32 and magnetic skyrmions 22,23 . The intricate interplay of higher-order and anisotropic bilinear magnetic interactions generates various magnetic states: conical spin spirals 33 and more complex magnetic structures 30,34,35 , such as an intricate nanoskyrmion lattice for a monolayer of Fe on the Ir(111) surface 36 .
In this work, we utilize a microscopic model combined with first-principles-based simulations to introduce and characterize a new kind of spin-orbit-driven magnetic pair interaction, the chiral biquadratic interaction (CBI). It has the form C ij · (S i × S j ) (S i · S j ). Like the DMI, this is a unidirectional interaction which is linear in the spin-orbit coupling, and so it is governed by the magnitude and orientation of the CBI vector C ij . We demonstrate that this vector obeys the same symmetry rules as the DMI 4,[37][38][39] . Like the isotropic biquadratic interaction, it couples twice a pair of magnetic moments. After systematic investigations on magnetic dimers made of 3d elements on various surfaces with strong spin-orbit coupling, namely Pt(111), Pt(001), Ir(111) and Re (0001) surfaces, we find that the CBI can be comparable in magnitude to the DMI. Lastly, we explore the implications of the CBI for magnetic structures in one and two dimensions.

Results
Systematic microscopic derivation of higher-order interactions The benefits of studying the properties of the magnetic interactions starting from a microscopic model are well-illustrated by the case of the DMI. Although phenomenological arguments completely determine the form and symmetry properties of the DMI 3 , the microscopic analysis of Moriya 4 and later on the intuitive picture proposed by Fert and Lévy 37,38 have clarified the main ingredients that underpin this inter-action. We thus begin by introducing a generic model of the electronic structure of the magnetic material, and then outline how one can systematically extract all kinds of magnetic interactions from the electronic grand potential.
Microscopic model. The microscopic hamiltonian that we consider has three contributions: H = is the local exchange coupling of strength U i between the magnetic moment S i on site i and the electronic spin σ, and H soc = a λ a L a · σ is the atomic spin-orbit coupling of strength λ a on site a between the electron spin and its atomic orbital angular momentum L a . Grouping the spindependent terms into ∆H = H mag + H soc , it is straightforward to derive a formal power series for the electronic grand potential (see Supplementary Note 1), Here Ω 0 is the contribution to the grand-canonical potential from the spin-independent H 0 , and   Fig. 1c and leads to the isotropic biquadratic interaction, The diagrams with the same number of lines but connecting either three or four different magnetic sites lead to the isotropic 4-spin 3-site and 4-spin 4-site interactions, respectively (see Supplementary Note 1). Lastly, we find a new kind of magnetic interaction from the prototypical diagram shown in Fig. 1d: We name it the chiral biquadratic interaction (CBI), as it is an antisymmetric 4-spin 2-site interaction generated by an additional spin-orbit site. It thus combines the isotropic scalar product S i · S j with the chiral coupling C ij · (S i × S j ) defined by the CBI vector C ij = a C ij,a , which is generated by the spin-orbit sites. This is our main quantity of interest and its properties will be discussed in detail in this paper. The diagrams with the same number of lines but connecting either three or four different magnetic sites lead to the chiral 4-spin 3-site and 4-spin 4-site interactions, respectively (see Supplementary Note 1).
Symmetry rules. We next study what are the properties of the newly-found CBI vector, C ij , by . The vanishing components of the DMI vector are those either parallel or perpendicular ton, which represents either the rotation axis or the normal to the mirror plane. It follows naturally from comparing the structure of the prototypical diagrams for the DMI and the CBI that precisely the same rules apply to the CBI vector, C ij . The two vectors do not have to be collinear, notably if the only applicable symmetry is of type (f).
Connection to a phenomenological model. Another advantage of of our approach is apparent if we consider the appropriate phenomenological model for the magnetic interactions. To illustrate this point, we consider the most general spin model containing only bilinear and biquadratic pair interactions: The bilinear interactions are described by a rank-2 cartesian tensor J αβ ij (9 parameters), which contains the isotropic pair interaction given by J ij S i · S j (1 parameter), the DMI given by D ij · E(α) = J cos α + D y sin α + B cos 2 α + C y sin α cos α .
Here α = θ 2 − θ 1 is the opening angle between the two magnetic moments. The angle that where the last line gives an approximation to the canting angle.
Magnetic dimers on Pt(111). We first compare the magnetic properties of five different homoatomic dimers on the Pt(111) surface, with the corresponding data collected in Table 1. All dimers except Ni possess large spin magnetic moments, which depend very weakly on the various imposed magnetic structures. Comparing the CBI to the DMI, we see that the magnitude of C y is around 20-30% of the one of D y , even reaching 60% for Ni. For most dimers, B is similar in magnitude to the CBI, and is even stronger than the DMI for Cr and Ni. According to J, which is the dominant interaction, Cr and Mn are antiferromagnetic, while Fe, Co and Ni are ferromagnetic. Considering only J and D y leads to a canting of the magnetic structure given by ∆α 2s in Table 1, while considering also B and C y we obtain ∆α 4s . The difference between these values is the largest for Cr and Fe, so these are the dimers for which the biquadratic interactions are  Fig. 2c   in Eq. (5)), but also to tilt in away from the direction defined by the DMI vector.
Cr and Fe dimers on other surfaces. The Cr and Fe dimers on Pt(111) were found to have the most important contributions from the CBI. To ascertain whether this is particular to the Pt(111) surface, we placed these dimers on other surfaces with strong spin-orbit coupling, namely Pt(001), Ir (111) and Re(0001). We see from Table 1 that the CBI is generally a sizeable fraction of the DMI. On the Pt(001) and Ir(111) surfaces, the two dimers display a very large DMI, even in relation to its isotropic bilinear interaction J, leading to a strong canting of the magnetic structure. This canting is substantially modified when the biquadratic interactions are accounted for. The same behavior is found for the Cr dimer on Re(0001), while for the Fe dimer on this surface the interactions are found to be surprisingly weak, but still support a strongly noncollinear magnetic structure.
Electronic origin of the magnetic interactions. The origin of the different magnetic interactions can be further understood by comparing their dependence on the filling of the electronic states with the corresponding density of states of each dimer. This is shown in Fig. 3 for the Cr and the Fe d-states. The prototypical diagram of Fig. 1b shows that the DMI is an interaction mediated by a spin-orbit site, which are supplied by the Pt surface atoms, and so this interaction is strongly dependent on the Pt d-states. The CBI involves both a direct exchange between the magnetic sites and an excursion through a spin-orbit site, Fig. 1d, so it can be amplified in those two ways, leading to a more complicated dependence.

Implications of the chiral biquadratic interaction The CBI has different important implications
for a broad class of noncollinear magnetic nanostructures. For a magnetic dimer, we already found that the CBI influences the opening angle and the vector spin chirality of the magnetic structure. To gain further understanding, we return to the previous example of a CBI vector in the y-direction, for which the interaction energy has the form E CBI (α) = C y sin α cos α (see Eq. (4)). The cos α term comes from theê 1 ·ê 2 part of the interaction, while the sin α terms comes from (ê 1 ×ê 2 ) y , with α the opening angle. As the dot product is isotropic, the opening is favored in the plane perpendicular to the CBI vector. Fixing C y > 0 for definiteness, there are two energy minima for α min ∈ {−45 • , 135 • }, and two maxima for α max ∈ {45 • , −135 • }. Strikingly, the two values of α min have opposite signs, which means that the sign of their vector spin chirality (projected on the y-axis) is also opposite. Thus, and in contrast to the DMI, the CBI favors both possible rotational senses at once (although with different opening angles). Starting from a ferromagnetic or antiferromagnetic structure (set by J), the DMI will induce a canting of the same rotational sense for both cases, while the CBI will favor cantings for each structure which have opposite rotational senses.
From a dimer to an infinite chain. Next we relate the magnetic ground state of a dimer to that of an infinite chain, assuming that the interactions present are J, D y and C y (we take B = 0 for simplicity), being nearest-neighbor interactions for the chain. The energy as a function of the opening angle for the dimer is given by Eq. (4), and the same form applies for the energy of a Impact of the CBI on complex 2D magnetic structures. As a final example, we consider twodimensional magnetic systems. Higher-order isotropic interactions can help stabilizing complex magnetic structures called multiple-Q-states [19][20][21]36 . We thus address the potential role of that the CBI might play for such complex magnetic structures, choosing the Fe monolayer on Ir(111) by way of example 36 . The ground state is a nanoskyrmion lattice, which is a type of 2Q-state made of two symmetry-related wavevectors Q 1 and Q 2 . Other combinations lead to further noncollinear states which were calculated to have a similar energy: single-Q spin spirals, the Q m -star and Q mvortex states, and the nanovortex lattice. These magnetic structures are visualized in Fig. 5a-

Discussion
We presented a comprehensive analysis of a new chiral higher-order magnetic pair interaction, the chiral biquadratic interaction (CBI). Using a microscopic model and a systematic expansion of the electronic grand-potential, we identified the prototypical diagrams behind all kinds of magnetic interactions. This led us to uncover a new chiral interaction, the CBI, which is linear in the spin-orbit coupling and is the biquadratic equivalent of the DMI, following the same symmetry rules. In its most general form, this interaction couples four distinct magnetic sites, and consists of terms of the form C ijkl · (S i × S j ) (S k · S l ). We note that a recent study has found signatures of higher-order interactions in magnetic chains on the Re(0001) surface 42 To determine the energy of a target magnetic configuration, the total energy functional is augmented by a Zeeman term enforcing the constraint, The constraining magnetic field is transverse to the orientation of the local magnetic moment, b i ·ê i = 0, and it opposes the magnetic force that acts on it if the magnetic structure is not a stationary point of the total energy functional, The induced moments in the surface atoms are allowed to relax without any constraint. The model parameters are then determined by linear least-squares fitting the constraining fields obtained for a set of self-consistent magnetic configurations to the form of the magnetic force supplied by the atomistic spin model. The magnetic configurations for the dimers have been chosen using a Lebedev grid 52 containing 14 directions for each atom, which is well-suited to describe spherical harmonics up to = 2, resulting in a total of 14 2 = 196 configurations, which using symmetry arguments (time-reversal invariance of the magnetic energy plus the spatial symmetries that apply on different surfaces) the number of configurations can be further reduced to 56 for (111) and (0001) surfaces and to 36 for (001) surfaces.