Dynamical and reversible control of topological spin textures

# Dynamical and reversible control of topological spin textures

E. A. Stepanov, C. Dutreix, M. I. Katsnelson Radboud University, Institute for Molecules and Materials, Heyendaalseweg 135, 6525AJ Nijmegen, Netherlands
Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France
###### Abstract

Recent observations of topological spin textures brought spintronics one step closer to new magnetic memories. Nevertheless, the existence of Skyrmions, as well as their stabilization, require very specific intrinsic magnetic properties which are usually fixed in magnets. Here we address the possibility to dynamically control their intrinsic magnetic interactions by varying the strength of a high-frequency laser field. It is shown that drastic changes can be induced in the antiferromagnetic exchange interactions and the latter can even be reversed to become ferromagnetic, provided the direct exchange is already non-negligible in equilibrium as predicted, for example, in Si doped with C, Sn, or Pb adatoms. In the presence of Dzyaloshinskii-Moriya interactions, this enables us to tune features of ferromagnetic Skyrmions such as their radius, making them easier to stabilize. Alternatively, such topological spin textures can occur in frustrated triangular lattices. Then, we demonstrate that a high-frequency laser field can induce dynamical frustration in antiferromagnets, where the degree of frustration can subsequently be tuned suitably to drive the material toward a Skyrmionic phase.

In the 1960s, Skyrme solved the equation of motion for a linear sigma model Lagrangian and reported static classical solutions which are now referred to as Skyrmions Skyrme (1962). Remarkably, the boundary conditions they satisfy allow them to be characterized by a topological charge. The elementary particles they described were identified as three-quark-made objects, namely, baryons, the family to which belong protons and neutrons. Skyrmions were later predicted in condensed matter physics too, as nontrivial spin textures Bogdanov and Yablonskii (1989). Importantly, this prediction has recently been confirmed experimentally by neutron scattering in three-dimensional helical magnets MnSi Mühlbauer et al. (2009) and FeCoSi Münzer et al. (2010), by electron microscopy in two-dimensional helical magnet FeCoSi Yu et al. (2010), and by spin-polarized scanning tunneling microscopy in Fe films deposited onto the Ir(111) surface  Heinze et al. (2011). The observations of such topological magnetic structures have been a decisive step forward in the perspective of Skyrmion-based data storage in spintronics Jonietz et al. (2010); Fert et al. (2013); Romming et al. (2013); Tomasello et al. (2014). From a fundamental viewpoint, skyrmions arise from different mechanisms. They appear in thin films under perpendicular magnetic field due to the competition between an easy-axis anisotropy and dipolar interactions that, respectively, favor out-of- and in-plane magnetizations Lin et al. (1973); Malozemoff and Slonczewski (1979); Garel and Doniach (1982). If they were also observed as a result of four-spin exchange interactions Heinze et al. (2011), this is within the context of frustrated exchange interactions (FEI)  Okubo et al. (2012); Leonov and Mostovoy (2015); Lin and Hayami (2016); Hayami et al. (2016) and Dzyaloshinskii-Moriya interactions (DMI) Dzyaloshinsky (1958); Moriya (1960); Pfleiderer (2011); Shibata et al. (2013); Emori et al. (2013); Ryu et al. (2013, 2014); Franken et al. (2014); Je et al. (2013); Hrabec et al. (2014); Chen et al. (2013) that Skyrmions are mainly discussed nowadays. In noncentrosymmetric ferromagnets DMI compete with the exchange interactions to yield a helical spiral phase which, under an external magnetic field, may lead to a Skyrmionic phase Han et al. (2010); Banerjee et al. (2014); Rosales et al. (2015); Keesman et al. (2015, 2016); Leonov et al. (2016). Nevertheless, DMI-based Skyrmions have a broad size, basically 5–100 nm, which makes them hard to stabilize Nagaosa and Tokura (2013); Banerjee et al. (2014). In frustrated magnets, FEI-based Skyrmions may also arise from the competition between ferromagnetic (FM) nearest-neighbor (NN) and antiferromagnetic (anti-FM) next-NN exchange interactions Okubo et al. (2012); Leonov and Mostovoy (2015); Lin and Hayami (2016); Hayami et al. (2016). However, the Skyrmionic phase additionally requires very special strengths for these two interactions. Thus, the main difficulty with controlling FEI- and DMI-based Skyrmions relies on the intrinsically fixed magnetic properties of materials. Tuning and controlling FEI and DMI then becomes extremely challenging. Research in this direction has recently been undertaken, thus reporting the possibility to tune DMI via anisotropy Emori et al. (2013); Ryu et al. (2013, 2014); Franken et al. (2014); Je et al. (2013); Hrabec et al. (2014); Chen et al. (2013), hydrostatic pressure Fåk et al. (2005); Ritz et al. (2013a, b), or mechanical strain Shibata et al. (2015).

Here, we report the possibility to dynamically control the intrinsic magnetic interactions by varying the strength of a high-frequency laser field, and subsequently tune the Skyrmionic features they are responsible for. The idea simply relies on the fact that DMI and FEI are both based on hopping processes, and that time-periodic fields renormalize the electronic tunneling, leading to phenomena such as dynamical Wannier-Stark localization Dunlap and Kenkre (1988), symmetry-protected topological transitions Oka and Aoki (2009); Kitagawa et al. (2010); Lindner et al. (2011); Carpentier et al. (2015); Dutreix et al. (2016); Cayssol et al. (2013), or ultrafast control of magnetism Takayoshi et al. (2014); Mikhaylovskiy et al. (2014); Mentink et al. (2015); Sato et al. (2016). Here, we show that drastic changes can be induced in the antiFM exchange interactions that can even be switched to FM, provided the direct exchange interaction is already reasonable in equilibrium. This dynamical anti-FM – FM phase transition is predicted in Si(111) doped with Sn or Pb adatoms under infrared light. Moreover, DMI are also renormalized by the laser field, which allows to dynamically tune features of FM Skyrmions such as their radius, making them easier to stabilize. In the case of FEI in triangular lattices, anti-FM Skyrmions have been predicted too, but no suitable magnets are available for experimental realizations so far. Then we suggest a possible route to induce dynamical frustration in antiferromagnets, and subsequently drive the degree of frustration until the material enters a Skyrmionic phase. Possible applications of this prescription are finally discussed in materials such as CF and Si(111) doped with C adatoms.

Skyrmion model with DMI – Let us start with the following tight-binding Hamiltonian

 H =∑⟨ij⟩,σσ′c∗iσ(tδσσ′+iΔijσσσ′)cjσ′+∑iU00ni↑ni↓ (1) +12∑⟨ij⟩,σσ′U⟨ij⟩niσnjσ′−12∑⟨ij⟩,σσ′JD⟨ij⟩c∗iσci,σ′c∗j,σ′cjσ,

where denotes the NN hopping amplitudes of electrons on a triangular lattice. Vector describes the Rashba spin orbit and lies perpendicularly to the bond between NN sites and , while is a vector of Pauli matrices. Besides, and refer to onsite and NN Coulomb interactions, and is the NN FM direct exchange interaction. The latter can be comparable to the anti-FM kinetic exchange interaction in LiCuO, SrCu(BO) and Si(111) with adatoms Mazurenko et al. (2007, 2008); Badrtdinov et al. (2016), and even compensate it in CMazurenko et al. (2016); Rudenko et al. (2013). The direct DMI is usually small and may even vanish in some 2D materials due to symmetry arguments Mazurenko et al. (2008); thus, it is disregarded here.

High-frequency description – Now we aim to provide an effective description of the system when electrons are rapidly driven by a time-periodic laser of frequency . The vector potential it leads to in the temporal gauge is , where is the lattice constant and . It is described via Peierls substitution in the momentum representation of the Hamiltonian, where denotes the electron charge. Phase characterizes the light polarization that is elliptic for and linear for . The Hamiltonian becomes time periodic and its quantum nonequilibrium steady states obey the time-dependent Schrödinger equation , where . Here, we have introduced a dimensionless parameter that compares a certain energy scale to the field frequency. For simplicity, we chose as the largest energy scale involved in Hamiltonian (1) among , , , and . Then, the Schrödinger equation reads , where the Hamiltonian is now renormalized as . In the high-frequency limit, is small and we can look for a unitary transformation defined as , which removes the time dependence of the Hamiltonian Itin and Neishtadt (2014); Itin and Katsnelson (2015). By construction we also impose , with a periodic function that averages at zero. Such a transformation leads to , where . Then and are determined iteratively in all orders in (see, e.g., Refs. Dutreix et al. (2016); Stepanov et al. ()). Here, we restrict the analysis to the second order in . The effective time-independent Hamiltonian it leads to is

 H =∑⟨ij⟩,σσ′c∗iσ(t′δσσ′+iΔ′ijσσσ′)cjσ′+∑iU00ni↑ni↓ (2) +12∑⟨ij⟩,σσ′U⟨ij⟩niσnjσ′−12∑ij,σσ′J′D⟨ij⟩c∗iσci,σ′c∗j,σ′cjσ.

Kinetic hopping and the Rashba spin orbit are both NN hopping processes, so they are both renormalized in the same way by the laser field: and , where is the 0th order Bessel function, the polarization is assumed to be circular (), and with the laser field strength. The explicit expression of the renormalized direct exchange interaction is provided in Ref. Stepanov et al. (). Importantly, the effective Hamiltonian derived above from the high-frequency expansion remains a good approximation of the dynamics over a time scale that is exponentially long with the frequency Kuwahara et al. (2016); Ho and Abanin (2016), and during which heating can be neglected. Indeed, the time scale after which the heating of the system becomes crucial is much larger than the measurement time . Here, is the number of driving periods  111The electron-electron interaction introduced in our model additionally plays the role of relaxation mechanism that allows to obtain such a nonequilibrium steady states Glazman (1981, 1983). The detailed investigation of the processes that help electrons to relax into such a stabilized regime is currently under investigations Asban and Rahav (2014); Raz et al. (2016); Seetharam et al. (2015); Canovi et al. (2016); Mori et al. (2016)..

In the strong localization regime (), one can construct a Heisenberg Hamiltonian in terms of spin operators and superexchange as proposed by Anderson Anderson (1959) and Moriya Moriya (1960)

 Hspin=−∑⟨ij⟩Jij^Si^Sj+∑⟨ij⟩Dij[^Si×^Sj]. (3)

Here, , where  Schüler et al. (2013). It characterizes DMI, namely, antisymmetric anisotropic interactions that are responsible for the weak ferromagnetism of some antiferromagnets Dzyaloshinsky (1958); Moriya (1960); Yildirim et al. (1995). This interaction scales with . Importantly, no additional contribution to DMI can be effectively induced by the high-frequency light Stepanov et al. () and, therefore, DMI cannot change signs when varying the field strength. Besides, there may be a third term in Eq. (3) which, as introduced in Moriya’s seminal paper Moriya (1960), describes symmetric anisotropic interactions. Nevertheless, it scales with and since , this term can safely be neglected for all strengths of the laser field Stepanov et al. (). Note that, finally, a Zeeman magnetic field could also be included in Hamiltonian (3) through as in Refs. Rosales et al. (2015); Hayami et al. (2016). However, it would neither be renormalized by the high-frequency field, nor be responsible for any correction up to the second order in the high-frequency expansion Stepanov et al. (). This is the reason why it is disregarded here. The isotropic symmetric exchange interaction between two spins satisfies . Here, denotes the direct exchange interactions which takes place in the material in equilibrium, i.e., in the absence of the laser field. The anti-FM kinetic exchange interaction already exists in equilibrium, but it is renormalized by the field strength. Finally is a FM field-induced correction to the direct exchange and is a purely nonequilibrium effect.

If the Hubbard Hamiltonian that leads to Eq. (3) lies in an anti-FM phase in equilibrium, it is remarkable that it undergoes a dynamical phase transition to become FM when varying the field strength out of equilibrium. This is illustrated by the positive values of exchange interaction in Fig. 1. Because of field-induced correction , this dynamical transition is even predicted to occur when the direct exchange is absent in equilibrium, as in iron oxides Dmitrienko et al. (2015). When , the transition roughly requires and, additionally, according to Fig. 1. Since  eV in iron oxides Dmitrienko et al. (2015), the laser strength of  eV/ involved at the transition would burn the material. A fortiori, reasonable strengths in iron oxides imply , so that corrections to are too small to induce the phase transition and can only yield negligible changes, in agreement with Refs. Mikhaylovskiy et al. (2014); Mentink et al. (2015).

Importantly, our work shows that the presence of exchange interaction in equilibrium is crucial to induce stronger changes in the exchange interaction with realistic laser strengths. Therefore, light control of magnetism looks more likely in p-block materials than in d-block transition metals. For example, the Si(111) surface doped with Pb or Sn adatoms is characterized by or  meV, or  meV, , or  meV and or  eV Badrtdinov et al. (2016), respectively. There, a laser field of frequency and strength  eV () would completely suppress the exchange interaction, thus inducing the anti-FM – FM phase transition dynamically, as shown in Fig. 1. Note that, if the direct exchange were null in equilibrium (), the situation would be similar to what happens in iron oxides.

The competition between exchange interaction and DMI may yield Skyrmions whose radius scales with  Bogdanov and Hubert (1994); Bocdanov and Hubert (1994); Han et al. (2010). Figure 1 shows that one can dynamically change this ratio by varying the laser strength. Thus, it becomes possible to engineer skyrmions of arbitrary small sizes, which usually makes them easier to stabilize in experiments. Importantly, with the absence of direct exchange DMI scales in the same way as exchange interaction and the ratio remains almost unchanged, which is again in agreement with the Refs. Mikhaylovskiy et al. (2014); Mentink et al. (2015). Skyrmion stabilization can be achieved under a perpendicular magnetic field. In the case of FM Skyrmions, this occurs for magnetic fields with a strength satisfying  Banerjee et al. (2014); Keesman et al. (2015). The stable Skyrmionic phase as a function of the laser field and magnetic field strengths is illustrated in Fig. 2, where the values of and are the ones obtained in Ref. Banerjee et al., 2014. The left-hand side of the plot shows that the high-frequency laser can help to stabilize Skyrmions by drastically enlarging the range of suitable magnetic fields. Anti-FM Skyrmions Zhang et al. (2016); Barker and Tretiakov (2016), however, are stabilized under high magnetic fields. For example, in Si(111):Pb, the Skyrmionic state was predicted to be stabilized under a 250 T magnetic field  Badrtdinov et al. (2016). Actually, it has been shown that the strength of the magnetic field scales linearly with the exchange interaction and in particular . Therefore, shining the material with a high-frequency laser may be relevant to significantly reduce the exchange interaction and, thus, to diminish the stabilizing magnetic field down to experimentally realistic strengths.

Finally, the possibility to undergo anti-FM – FM phase transition by varying the laser strength allows one to generate two different types of Skyrmions in one system and observe the anti-FM Skyrmionic – FM Skyrmionic phase transition. Indeed, one can stabilize the anti-FM Skyrmions, for example, obtained in Si(111):Pb Badrtdinov et al. (2016), under the influence of the high-frequency light by applying a weak perpendicular magnetic field in the antiferromagnetic phase. After driving the system toward the FM phase the exchange interaction changes sign and one can then stabilize the new FM Skyrmionic structure by adjusting the magnetic field.

skyrmion model – Now let us consider another interesting model that describes a frustrated magnetic system. It consists of an isotropic Heisenberg model on a triangular lattice where Skyrmions appear as a result of the competition between strong ferromagnetic NN and weak antiferromagnetic next-NN exchange interactions. In order to obtain Skyrmions, these interactions must obey special conditions that originate from the lattice structure Okubo et al. (2012); Leonov and Mostovoy (2015); Lin and Hayami (2016); Hayami et al. (2016). Thus, for the model the exchange interactions should satisfy , whereas they should satisfy in the model. Designing such a frustrated system is of course a nontrivial problem experimentally. Nevertheless, we subsequently show that frustration can be realized by shinning an antiferromagnet with a high-frequency laser.

We consider the single-band extended Hubbard Hamiltonian (1) on a triangular lattice but with NN and next-NN hopping processes and Coulomb interactions

 (4)

Using the high-frequency expansion introduced above, one can obtain an effective Hamiltonian which, for circularly polarized fields, is

 (5)

where the renormalized hopping amplitudes are , , and . The explicit expression of the renormalized exchange interaction is detailed in Ref. Stepanov et al. ().

When the system lies in the strong interaction regime, one can write an effective Heisenberg model,

 (6)

with NN exchange interaction and next-NN exchange interaction . Top right panel in Fig. 1 shows that, for vanishing laser fields, the system lies in the antiferromagnetic phase. When turning on the laser field and increasing its strength, the system undergoes a transition toward a ferromagnetic phase. Importantly, the nearest-neighbor and the next-NN exchange interactions, namely, and , become ferromagnetic for different values of the field, meaning that one can engineer a frustrated magnet. Here we took the “unrealistic” case of just to make the anti-FM – FM transition more visible in the figure. Fig. 3 shows the phase diagram based on conditions and , as a function of and laser strength . Thus, the initial antiferromagnet may be dynamically driven toward the frustrated magnetic system predicted in Ref. Okubo et al., 2012 with suitable values of anti-FM and FM exchange interactions to obtain Skyrmions. In the case of Si(111) with C adatoms, it is estimated that  meV,  meV,  meV,  eV,  eV Badrtdinov et al. (2016), so that fields with frequency  eV and amplitudes would induce suitable values of to obtain Skyrmions, according to the left panel in Fig. 3. Similar effects are predicted in CF, where  meV,  meV,  meV and  eV,  eV Mazurenko et al. (2016) (see right panel of Fig. 3).

So far we have only considered the case of a circular polarization. For example, in the case of the square lattice under the influence of the noncircular polarized fields, the hopping amplitude and spin-orbit coupling vector are renormalized by the Bessel functions , where the labels correspond to the direction of the vector that connects two lattice sites. This allows us to change DMI and the Skyrmion radius in an anisotropic way. This case was recently investigated in Ref. Shibata et al., 2015, where DMI are tuned by strain forces, which changes the Skyrmion shape from circular to elliptic.

To summarize, we have reported the possibility to dynamically control the intrinsic magnetic interactions of two-dimensional materials. This can induce drastic changes in the anti-FM exchange interaction that can even be switched to FM, provided the direct exchange interaction in equilibrium is non-negligible. Additionally, the high-frequency laser field also renormalizes the DMI, so that Skyrmion features such that their radius can be tuned too, thus making them easier to stabilize under perpendicular magnetic fields. Besides, it has been shown that a high-frequency laser field can also induce dynamical frustration in antiferromagnetic triangular lattices, where the degree of frustration can be tuned suitably to experience Skyrmions. Importantly, the dynamical effects we have discussed within the high-frequency limit rely on laser strengths and frequencies that remain reasonable for realizations in solid state physics. In particular, we expect them to be relevant when irradiating sp and p materials like CF and Si(111):{C, Sn, Pb}.

###### Acknowledgements.
The authors thank A. N. Rudenko, V. V. Mazurenko, T. Kuwahara and A. Kimel for fruitful discussions and comments. This work was supported by NWO via Spinoza Prize and by ERC Advanced Grant 338957 FEMTO/NANO. Also, E. A. S. and M. I. K. acknowledge the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).

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## Appendix A Supplemental Material for “Dynamical and reversible control of topological spin textures”

E. A. Stepanov, C. Dutreix, M. I. Katsnelson

Radboud University, Institute for Molecules and Materials, Heyendaalseweg 135, 6525AJ Nijmegen, Netherlands

Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France

### a.1 Fourier transform of the kinetic part of the Hubbard-like Hamiltonian with DMI

The Fourier transform of the time-dependent part of the Hamiltonian that accounts for the effects of the high-frequency laser field can be obtained with the use of the followgin well-known relations which arise from the definition of the Bessel function of the -th order :

 ∫+π−πdt2πe−iZysin(t−ϕ)eimt=∫+π−πdt2πe−iZysint′eim(t′+ϕ)=eimϕJm(Zy), ∫+π−πdt2πe−iZxcost−iZysin(t−ϕ)eimt=eimθJm√Z2x+Z2y−2ZxZysinϕ,

where . This results in the following expression for the Fourier transform of the hopping amplitude and the spin-orbit coupling

 εk,m =2tJm(Z)[cos(kx−mπ/2)+cos(kx/2+ky√3/2)eim5π/6+cos(kx/2−ky√3/2)eimπ/6], fxk,m =√3iΔJm(Z)[sin(kx/2−ky√3/2)eimπ/6−sin(kx/2+ky√3/2)eim5π/6], fyk,m =iΔJm(Z)[2sin(kx−mπ/2)+sin(kx/2−ky√3/2)eimπ/6+sin(kx/2+ky√3/2)eim5π/6], (7)

where we consider the case of the circularly polarized light (, ) for simplicity. Then, the full Hamiltonian in the frequency space can be written as

 Hm= ∑k,σσ′c∗kσ(εk,mδσσ′+ifk,mσσσ′)ckσ′+Vδm0, (8)

where the time-independent interaction term transforms as .

### a.2 High-frequency description

As it was mentioned in the main text, the time-periodic Hamiltonian obey the time-dependent Schrödinger equation

 i∂τΨ(λ,τ)= 1ΩH(τ)Ψ(λ,τ), (9)

where . One can introduce a dimensionless parameter which compares a certain energy scale to the typical field energy. For simplicity we chose as the largest characteristic energy involved in the initial Hamiltonian among , and , so that no resonant processes with the applied laser field will occur. Then, the Schrödinger equation can be rewritten as

 i∂τΨ(λ,τ)= λ¯¯¯¯¯H(τ)Ψ(λ,τ), (10)

where the Hamiltonian is renormalized on the same energy scale as

 1ΩH(τ)=λH(τ)δE=λ¯¯¯¯¯H(τ). (11)

In order to obtain the effective Hamiltonian of our model we look for a unitary transformation defined as

 Ψ(λ,τ)=exp{−iΔ(τ)}ψ(λ,τ), (12)

which would remove the time dependence of the Hamiltonian. By construction we also impose

 Δ(τ)= +∞∑n=1λnΔn(τ) (13)

with a periodic function that averages at zero. Such a transformation leads to

 i∂τψ(λ,τ)=1ΩHψ(λ,τ)=λ¯¯¯¯¯Hψ(λ,τ) (14)

with effective Hamiltonian

 ¯¯¯¯¯H=eiΔ(τ)¯¯¯¯¯H(τ)e−iΔ(τ)−iλ−1eiΔ(τ)∂τe−iΔ(τ) , (15)

or equivalently

 H=eiΔ(τ)H(τ)e−iΔ(τ)−iΩeiΔ(τ)∂τe−iΔ(τ) . (16)

The partial time-derivative satisfies the following relation

 ∂τe−iΔ(τ) =∞∑n=0{(−iΔ(τ))n,−i∂τΔ(τ)}(n+1)! e−iΔ(τ) , (17)

where the repeated commutator is defined for two operators and by and . The square brackets denote the usual commutator. Then, one can write

 ¯¯¯¯¯H=eiΔ(τ)[¯¯¯¯¯H(τ)−iλ−1∞∑n=0{(−iΔ(τ))n,−i∂τΔ(τ)}(n+1)!]e−iΔ(τ) , (18)

Using the series representation

 ¯¯¯¯¯H=∞∑n=0λn~Hn, (19)

together with Eqs. (17) and (15), one can then determine operators and iteratively in all orders in . Here, we restrict ourselves to the case of the high-frequency laser field, which allows us to consider the effective Hamiltonian representation up to the second order correction in : . These effective time-independent Hamiltonians describe the stroboscopic dynamics of the system, whereas its evolution between two stroboscopic times is encoded into the time-dependent function  Dutreix et al. (2016).

As it was showed in the Ref. Dutreix et al. (2016), the first term in this representation is given by the time-average , where

 ¯¯¯¯¯Hm=∫+π−πdτ2π eimτ¯¯¯¯¯H(τ). (20)

Since in the initial problem the interaction term was time-independent, the time-averaging procedure changes only the single-particle terms of the Hamiltonian , which results in the renormalization of the hopping amplitude and spin-orbit coupling with respect to the time-independent problem as

 H≃~H0δE=∑⟨ij⟩,σσ′c∗iσ(t′δσσ′+iΔ′ijσσσ′)cjσ′+∑iU00ni↑ni↓+12∑⟨ij⟩,σσ′U⟨ij⟩niσnjσ′−12∑⟨ij⟩,σσ′JD⟨ij⟩c∗iσciσ′c∗jσ′cjσ,

where

 t′=tJ0(Z)  and  Δ′=ΔJ0(Z). (21)

### a.3 First-order correction ~H1 to the time-averaged Hamiltonian in the high-frequency expansion

The first-order in term in the effective Hamiltonian is given by the following equation

 ~H1=−∑m>0[¯¯¯¯¯Hm,¯¯¯¯¯H−m]m. (22)

Since does not contribute to the first-order term , one can rewrite the time dependent part of the Hamiltonian as follows

 ¯¯¯¯¯Hm>0= ∑k¯¯¯εk,m(c∗k↑ck↑+c∗k↓ck↓)+i∑k¯¯¯fxk,m(c∗k↑ck↓+c∗k↓ck↑)+∑k¯¯¯fyk,m(c∗k↑ck↓−c∗k↓ck↑), (23)

where we use the same notations for the renormalized variables and .
Let us compute the general commutator () that in our case splits into the three different terms

 ¯¯¯εk1,m¯¯¯εk2,n [(c∗k1↑ck1↑+c∗k1↓ck1↓),(c∗k2↑ck2↑+c∗k2↓ck2↓)], (24a) ¯¯¯εk1,m¯¯¯fik2,n [(c∗k1↑ck1↑+c∗k1↓ck1↓),(c∗k2↑ck2↓∓c∗k2↓ck2↑)], (24b) ¯¯¯fik1,m¯¯¯fjk2,n [(c∗k1↑ck1↓∓c∗k1↓ck1↑),(c∗k2↑ck2↓∓c∗k2↓ck2↑)]. (24c)

Let us start from Eq. (24a). Using the commutation relations one can obtain that

 c∗k1↑ck1↑c∗k2↑ck2↑ =−c∗k1↑c∗k2↑ck1↑ck2↑+δk1,k2c∗k1↑ck2↑=−c∗k2↑c∗k1↑ck2↑ck1↑+δk1,k2c∗k1↑ck2↑ =−c∗k2↑ck2↑c∗k1↑ck1↑−δk1,k2c∗k2↑ck1↑+δk1,k2c∗k1↑ck2↑=c∗k2↑ck2↑c∗k1↑ck1↑, c∗k1↓ck1↓c∗k2↑ck2↑ =−c∗k2↑ck2↑c∗k1↓ck1↓. (25)

Making the same transformations with other remaining terms, one can see that commutator in Eq. (24a) is equal to zero.
The result of Eq. (24b) can be obtained in the same style.

 c∗k1↑ck1↑c∗k2↑ck2↓+c∗k1↓ck1↓c∗k2↑ck2↓ =−c∗k1↑c∗k2↑ck1↑ck2↓+δk1k2c∗k1↑ck2↓−c∗k1↓c∗k2↑ck1↓ck2↓ (26) =−c∗k2↑c∗k1↑ck2↓ck1↑+δk1k2c∗k1↑ck2↓−c∗k2↑c∗k1↓ck2↓ck1↓ =−c∗k2↑ck2↓c∗k1↑ck1↑+δk1k2c∗k1↑ck2↓+c∗k2↑ck2↓c∗k1↓ck1↓−δk1k2c∗k2↑ck1↓ =−c∗k2↑ck2↓c∗k1↑ck1↑+c∗k2↑ck2↓c∗k1↓ck1↓.

Obtaining the second term with the similar transformations, one obtains that commutator in Eq. (24b) is also equal to zero.
The last commutator given by Eq. (24c) is zero as well. Indeed, since

 c∗k1↓ck1↑c∗k2↑ck2↓+c∗k1↑ck1↓c∗k2↓ck2↑ =−c∗k1↓c∗k2↑ck1↑ck2↓+δk1k2c∗k1↓ck2↓−c∗k1↑c∗k2↓ck1↓ck2↑+δk1k2c∗k1↑ck2↑ (27) =−c∗k2↑c∗k1↓ck2↓ck1↑+δk1k2c∗k1↓ck2↓−c∗k2↓c∗k1↑ck2↑ck1↓+δk1k2c∗k1↑ck2↑ =−c∗k2↑ck2↓c∗k1↓ck1↑−δk1k2c∗k2↑ck1↑+δk1k2c∗k1↓ck2↓+c∗k2↓ck2↑c∗k1↑ck1↓−δk1k2c∗k2↓ck1↓+δk1k2c∗k1↑ck2↑ =−c∗k2↑ck2↓c∗k1↓ck1↑+c∗k2↓ck2↑c∗k1↑ck1↓, c∗k1↑ck1↓c∗k2↑ck2↓ =−c∗k2↑ck2↓c∗k1↑ck1↓, (28)

the commutator in Eq. (24c) is equal to zero. So, if for all , the first-order correction term of the effective Hamiltonian is identically zero.

### a.4 Second-order term ~H2 of the Hamiltonian in the high-frequency expansion

Since as it was shown above, the second-order term of the effective Hamiltonian can be simplified as

 ~H2=∑m>0[[¯¯¯¯¯Hm,¯¯¯¯¯H0],¯¯¯¯¯H−m]m2=∑m>0[[¯¯¯¯¯Hm,¯¯¯¯V],¯¯¯¯¯H−m]m2. (29)

It should be mentioned, that we consider the high-frequency description of the effective Hamiltonian until the second order in . In general, one can stop at the first-order term , because the second order term describes only the corrections to the interactions, that will be times smaller than the interactions already described by the time averaged term . Therefore, they are negligibly small and a priori not important for the magnetic properties of the system. Nevertheless, according to the Ref. Itin and Katsnelson (2015), the high-frequency field generates new types of interaction terms that were not present in the initial problem. In particular, there is one important correction, namely , to the direct exchange interaction term that comes from the Coulomb interaction and appears exactly in the second-order correction in the effective Hamiltonian. After taking th