Ultrafast Control of Spin Interactions in Honeycomb Antiferromagnetic Insulators

TL;DR

This paper demonstrates how polarized laser pulses can ultrafastly and nonthermally control exchange and Dzyaloshinskii-Moriya interactions in honeycomb antiferromagnetic insulators, enabling rapid spin manipulation.

Contribution

It introduces a method to tune spin interactions in 2D honeycomb lattices using Floquet theory, revealing control over their amplitudes, signs, and ratios with laser parameters.

Findings

Laser pulses can modify spin interaction strengths and signs.

Control over exchange and Dzyaloshinskii-Moriya interactions achieved.

Renormalizations are independent of laser helicity.

Abstract

Light enables ultrafast, direct and nonthermal control of the exchange and Dzyaloshinskii-Moriya interactions. We consider two-dimensional honeycomb lattices described by the Kane-Mele-Hubbard model at half filling and in the strongly correlated limit, i.e., the Mott insulator phase of a canted antiferromagnet. Based on Floquet theory, we demonstrate that by changing the amplitude and frequency of polarized laser pulses, one can tune the amplitudes and signs of and even the ratio between the exchange and Dzyaloshinskii-Moriya spin interactions. Furthermore, the renormalizations of the spin interactions are independent of helicity. Our results pave the way for ultrafast optical spin manipulation in recently discovered two-dimensional magnetic materials.

Figures

Equations

$${\hat{H}}_0={\hat{\mathcal{H}}}_{\text{K}}+{\hat{\mathcal{H}}}_{\text{SOI}}+{\hat{\mathcal{H}}}_{\text{int}},$$

$${\hat{\mathcal{H}}}_{\text{K}}=-t_1\sum _{⟨i,j⟩,\tau }{\hat{c}}_{i\tau }^{†}{\hat{c}}_{j\tau }-t_2\sum _{⟨⟨i,j⟩⟩,\tau }{\hat{c}}_{i\tau }^{†}{\hat{c}}_{j\tau },$$

$${\hat{\mathcal{H}}}_{\text{SOI}}=i\Delta \sum _{⟨⟨i,j⟩⟩,\tau ,{\tau }^{\prime }}{\nu }_{ij}{\sigma }_{\tau ,{\tau }^{\prime }}^z{\hat{c}}_{i\tau }^{†}{\hat{c}}_{j{\tau }^{\prime }},$$

$${\hat{\mathcal{H}}}_{\text{int}}=U_{00}\sum _{i=1}{\hat{n}}_{i\uparrow }{\hat{n}}_{i\downarrow }+\frac{1}{2}\sum _{⟨i,j⟩,\tau {\tau }^{\prime }}{V}_{ij}{\hat{n}}_{i\tau }{\hat{n}}_{j{\tau }^{\prime }}.$$

$${\hat{\mathcal{H}}}_{\text{int}}\approx U\hat{d},$$

$${\hat{c}}_{i\tau }^{†}{\hat{c}}_{i{\tau }^{\prime }}=\frac{1}{2}(n_{i\uparrow }+n_{i\downarrow }){\delta }_{\tau {\tau }^{\prime }}+{\mathbf{S}}_i\cdot {\mathbf{\tau }}_{{\tau }^{\prime },\tau },$$

$${\hat{c}}_{i\tau }{\hat{c}}_{i{\tau }^{\prime }}^{†}=\frac{1}{2}(2-n_{i\uparrow }-n_{i\downarrow }){\delta }_{\tau {\tau }^{\prime }}-{\mathbf{S}}_i\cdot {\mathbf{\tau }}_{\tau ,{\tau }^{\prime }},$$

$$H_{\text{S}}=$$

$$+\sum _{⟨⟨i,j⟩⟩}{\mathbf{S}}_i\mathbf{\Gamma }{\mathbf{S}}_j+\sum _{⟨⟨i,j⟩⟩}{\mathbf{D}}_{ij}\cdot {\mathbf{S}}_i\times {\mathbf{S}}_j,$$

$$J_{1(2)}=\frac{2t_{1(2)}^2}{U},$$

$$\mathbf{\Gamma }=\frac{2{\Delta }^2}{U}\text{diag}(-1,-1,1),$$

$${\mathbf{D}}_{ij}=\frac{4t_2\Delta }{U}{\nu }_{ij}{\hat{\mathrm{e}}}_z.$$

$$e^{i\frac{e}{\hslash }{\mathbf{R}}_{ij}\cdot \mathbf{A}(t)}=\sum _m{e}^{i(\frac{\pi }{2}-{\theta }_{ij})m}{\mathcal{J}}_m({\alpha }_{ij})e^{im\omega t},$$

$$\hat{T}(t)=\sum _m({\hat{T}}_{-1,m}+{\hat{T}}_{0,m}+{\hat{T}}_{1,m})e^{im\omega t}.$$

$${\hat{H}}^{\prime }(t)=e^{i\hat{S}(t)}[\hat{H}(t)-i{\partial }_t]e^{-i\hat{S}(t)}.$$

$${\hat{S}}^{(\nu )}(t)=\sum _{d\ne 0}\sum _m{\eta }^{\nu }{\hat{S}}_{d,m}^{(\nu )}{e}^{im\omega t},$$

$${\hat{T}}^{\prime }(t)\approx -\sum _m{\hat{T}}_{0,m}(t)e^{im\omega t}$$

$$+\frac{1}{2}\sum _{mn}\left(\frac{\left[{\hat{T}}_{1,n},{\hat{T}}_{-1,m-n}\right]}{U+n\hslash \omega }-\frac{\left[{\hat{T}}_{-1,n},{\hat{T}}_{1,m-n}\right]}{U-n\hslash \omega }\right)e^{im\omega t}.$$

$${\hat{H}}_{\text{eff}}$$

$${\tilde{H}}_{\text{S}}(\omega )=$$

$$+\sum _{⟨⟨i,j⟩⟩}{\mathbf{S}}_i{\tilde{\mathbf{\Gamma }}}_{ij}{\mathbf{S}}_j+\sum _{⟨⟨i,j⟩⟩}{\tilde{\mathbf{D}}}_{ij}\cdot {\mathbf{S}}_i\times {\mathbf{S}}_j,$$

$${\tilde{J}}_{1,ij}=2t_1^2\sum _n\frac{{\mathcal{J}}_n^2({\alpha }_{⟨ij⟩})}{U+n\hslash \omega },$$

$${\tilde{J}}_{2,ij}=2t_2^2\sum _n\frac{{\mathcal{J}}_n^2({\alpha }_{⟨⟨ij⟩⟩})}{U+n\hslash \omega },$$

$${\tilde{\mathbf{\Gamma }}}_{ij}=2{\Delta }^2\text{diag}(-1,-1,1)\sum _n\frac{{\mathcal{J}}_n^2({\alpha }_{⟨⟨ij⟩⟩})}{U+n\hslash \omega },$$

$${\tilde{\mathbf{D}}}_{ij}=4t_2\Delta \sum _n\frac{{\mathcal{J}}_n^2({\alpha }_{⟨⟨ij⟩⟩})}{U+n\hslash \omega }{\nu }_{ij}{\hat{\mathrm{e}}}_z.$$

Full text

Introduction. The discovery of the all-optical control of the order parameters in antiferromagnetic (AFM) and ferromagnetic (FM) materials by means of ultrashort intense high-frequency laser pulses has propelled spintronics into a new era of ultrafast magnetism Kimel et al. (2005, 2009); Kirilyuk et al. (2010). Despite the many attempts to uncover its origin, the detailed underlying microscopic mechanism remains unclear. The process triggering ultrafast magnetization dynamics phenomena may rely on either thermal or nonthermal mechanisms Ostler et al. (2012); Kirilyuk et al. (2010). Thus far, it has been believed that the ultrafast nonthermal manipulation of spins is possible only through either a direct coupling between the magnetic field component of the laser pulses and the spins or an indirect coupling between the electric field component of the light and the spins via spin-orbit coupling Kirilyuk et al. (2010); Mikhaylovskiy et al. (2015); Qaiumzadeh et al. (2013); Qaiumzadeh and Titov (2016). Recent experiments have demonstrated that laser pulses can directly modify the amplitude and sign of the exchange interaction, which is the strongest spin interaction in magnetically ordered systems Görg et al. (2018); Mikhaylovskiy et al. (2015); Li et al. (2013); Ju et al. (2004). References Mentink and Eckstein, 2014; Itin and Katsnelson, 2015; Mentink et al., 2015; Meyer et al., 2017; Stepanov et al., 2017; Kitamura et al., 2017; Mentink, 2017; Barbeau et al., 2019 have theoretically proposed that a direct coupling between the electric field of the light and the spins facilitates the nonthermal optical modification of the exchange interaction, in agreement with recent experiments Görg et al. (2018); Mikhaylovskiy et al. (2015).

In magnetic systems with broken inversion symmetry, there is also an antisymmetric exchange interaction between spins that breaks the chiral symmetry, namely, the Dzyaloshinskii-Moriya interaction (DMI) Dzyaloshinsky (1958); Moriya (1960); Qaiumzadeh et al. (2018a); Ado et al. (2018). Although this interaction is considerably weaker than the exchange interaction, it is essential in magnetic materials for enabling weak ferromagnetism in AFM materials Dzyaloshinsky (1958); Moriya (1960), topological objects such as chiral skyrmions Mühlbauer et al. (2009); Fert et al. (2017); Flovik et al. (2017); Khoshlahni et al. (2019) and chiral domain walls Thiaville et al. (2012); Ryu et al. (2013); Qaiumzadeh et al. (2018b), and exotic phases of topological magnon insulators Owerre (2016); Kim et al. (2016); Elyasi et al. (2019); Chen et al. (2018); Owerre et al. (2019); Owerre . The ratio between the exchange interaction and the DMI controls the tilt angle of the canted spins. Finding a mechanism for tuning this ratio can enable new phenomena in ultrafast spin dynamics and switching Mikhaylovskiy et al. (2015); Mentink et al. (2015); Stepanov et al. (2017); Kitamura et al. (2017).

Another far-reaching recent breakthrough in spintronics is the discovery of two-dimensional (2D) van der Waals AFM and FM materials with metallic, semiconducting and insulating band structures Mounet et al. (2018); Feng et al. (2017). The advantages presented by the existence of low dimensionality and magnetic order in the same material enable the development of new spintronic devices with exceptional performance.

In this Rapid Communication, we show that intense high-frequency laser pulses can dramatically affect the spin-spin interactions and the ratio between the exchange interaction and the DMI in a broad class of 2D magnetic materials described by the Kane-Mele-Hubbard model. We find that light can be used to tune both the sign and magnitude of the AFM exchange interaction, in agreement with the dynamical mean-field theory Mentink et al. (2015). Importantly, we demonstrate that laser pulses can also be used to independently change the sign and magnitude of the DMI, thus enabling the rapid control of the magnetism. The ability to independently control the signs and magnitudes of the exchange interaction and the DMI enables superior control of magnetic textures in 2D magnets.

Model Hamiltonian. The electron dynamics in 2D planar honeycomb lattices can be described by the Kane-Mele-Hubbard model Kane and Mele (2005); Yu et al. (2011); Hohenadler et al. (2011); Rachel and Le Hur (2010); Wen et al. (2011); Griset and Xu (2012); Auerbach (1994); see Fig. 1. In the absence of external perturbations, the Hamiltonian is the sum of the kinetic term $\hat{\mathcal{H}}_{\text{K}}$, the intrinsic spin-orbit interaction (SOI) $\hat{\mathcal{H}}_{\text{SOI}}$, and the repulsive Coulomb interaction between the electrons as modeled in the form of the extended Hubbard interaction $\hat{\mathcal{H}}_{\text{int}}$,

[TABLE]

where

[TABLE]

Here, $\langle\cdot\rangle$ and $\langle\langle\cdot\rangle\rangle$ denote nearest neighbors (NN) and next-nearest neighbors (NNN), respectively; $\hat{c}_{i\tau}^{\dagger}$ and $\hat{c}_{i\tau}$ are the fermionic creation and annihilation operators, respectively, for an electron at site $i$ and in spin state $\tau=\{\uparrow,\downarrow\}$; $t_{1}$ and $t_{2}$ are the NN and NNN hopping amplitudes, respectively; $\Delta$ is the intrinsic SOI parameter; $\nu_{ij}=\pm 1$, depending on the hopping orientation from $j$ to $i$ (see Fig. 1); $\sigma^{z}$ is the $z$ component of the Pauli matrices $\bm{\sigma}$; and $U_{00}$ and $V_{ij}$ are the on-site and NN Coulomb interactions, respectively. The intrinsic NNN SOI, Eq. (3), reduces the $\mathrm{SU}(2)$ symmetry of the original Hubbard model to the $\mathrm{U}(1)$ spin group. In buckled noncoplanar honeycomb lattices or systems with structural inversion asymmetry, the presence of NNN or NN Rashba SOIs, respectively, further reduces the symmetry to $Z_{2}$.

Using the variational principle, it has been shown that the NN Coulomb interaction can be approximated by a renormalized local interaction $U=U_{00}-\bar{V}$, where $\bar{V}$ is a weighted average of the NN Coulomb interaction Schüler et al. (2013). Thus, we consider only the local Coulomb interaction in our total Hamiltonian and express the interaction part of the Hamiltonian in Eq. (4) as

[TABLE]

where we introduce the doublon number operator $\hat{d}=\sum_{i=1}\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}$, which has an eigenvalue of $d$. For later use, we define $\hat{P}_{d}$ as the projection operator onto the subspace spanned by states with eigenvalue $d$, i.e., states with exactly $d$ doublons. At site $i$, we can define the projection operator related to double occupancy as $\hat{P}_{i,1}=\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}$ and the projection operator related to the absence of double occupancy as $\hat{P}_{i,0}=1-\hat{P}_{i,1}$ Fazekas (1999). We can then define the projection operator $\hat{P}_{d}$ for the whole system as follows. Let $\mathcal{O}$ and $\mathcal{P}^{d}(\mathcal{O})$ denote the set of sites on the lattice and the set of subsets of $\mathcal{O}$ with exactly $d$ elements, respectively. Then, in compact form, the projection operator reads $\hat{P}_{d}=\sum_{A\in\mathcal{P}^{d}(\mathcal{O})}\left\{\prod_{i\in A}\hat{P}_{i,1}\prod_{i\notin A}\hat{P}_{i,0}\right\}$.

We are interested in the strongly correlated regime $U\gg t_{1(2)}$ at half filling. In the limit of such strong coupling, any state with a nonzero number of double occupancies ($d\neq 0$) has a much larger energy than a state with no double occupancy ($d=0$). We obtain the effective Hamiltonian acting on the $d=0$ subspace by means of applying second-order perturbation theory on the hopping terms. Using the relations

[TABLE]

we find the spin Hamiltonian for 2D AFM spin-orbit Mott insulators,

[TABLE]

with the following spin-spin interactions,

[TABLE]

In the spin Hamiltonian given in Eq. (Ultrafast control of spin interactions in honeycomb antiferromagnetic insulators), the first and second terms are the NN and NNN symmetric Heisenberg AFM exchange interactions ($J_{1(2)}>0$), respectively; the third term is the NNN anisotropic exchange interaction (an XXZ-like term) arising from the intrinsic SOI, and the last term is the intrinsic NNN DMI. The intrinsic SOI in the Kane-Mele-Hubbard model, Eq. (3), leads to an NNN DMI with a DM vector $\bm{D}_{ij}$, which is perpendicular to the honeycomb layer with an amplitude linearly proportional to the SOI strength. It can also be shown that the breaking of the inversion symmetry in this system induces a Rashba-type SOI, which consequently results in an NN interfacial DMI with a DM vector that lies in the film plane and normal to the lattice bonds.

Although the microscopic derivation of the anisotropic exchange interaction in the spin Hamiltonian of Eq. (Ultrafast control of spin interactions in honeycomb antiferromagnetic insulators) has been reported before Rachel and Le Hur (2010); Vaezi et al. (2012); Lado and Fernández-Rossier (2017), we are not aware of any other microscopic calculation of the intrinsic NNN DMI in honeycomb lattices Owerre (2016); Kim et al. (2016); Chen et al. (2018). The spin Hamiltonian of Eq. (Ultrafast control of spin interactions in honeycomb antiferromagnetic insulators) gives rise to several interesting features and exotic phases, such as the existence of the magnon spin Nernst effect in collinear AFM layers Cheng et al. (2016); Zyuzin and Kovalev (2016), a topological magnon insulator phase Owerre (2016); Kim et al. (2016); Elyasi et al. (2019); Chen et al. (2018), spin Hall effects for Weyl magnons Zyuzin and Kovalev (2018); Sekine and Nomura (2016), magnonic Floquet topological insulators, spin density waves Mulder et al. (2010), and chiral and topological gapped spin liquid phases Vaezi et al. (2012). The ultrafast control of the DMI and the exchange interaction, by means of laser pulses, can enable the engineering of all of these phenomena and phases.

For completeness, let us briefly illustrate the effect of disorder by adding an onsite disorder potential $\sum_{i\tau}\varepsilon_{i}\hat{c}_{i\tau}^{\dagger}\hat{c}_{i\tau}$ to the Kane-Mele-Hubbard Hamiltonian given in Eq. (1), where $\varepsilon_{i}$ is an uncorrelated random variable. Following the above procedure, it can be shown that the spin interaction parameters are renormalized as $1/U\rightarrow 1/[U-(\epsilon_{j}-\epsilon_{i})^{2}/U]$ Protopopov and Abanin (2019). In the large $U$ limit and in the presence of very high frequency oscillations, the effect of disorder is negligible; thus, we do not include it in the following.

Laser illumination. We introduce the effects of laser irradiation in the Kane-Mele-Hubbard Hamiltonian of Eq. (1) via the Peierls substitution Peierls (1933). The Peierls’ prescription is valid for slowly varying vector potentials on the scale of the lattice constant at which the system remains in a quasiequilibrium state Luttinger (1951). The electric field component of a polarized laser pulse is given by $\bm{E}(t)=E_{0}(e^{-i\omega t}\hat{\epsilon}+\mathrm{c.c.})/2$, where $E_{0}$ is the electric field amplitude; $\omega$ is the laser pulse frequency; and $\hat{\epsilon}=(\hat{\mathrm{e}}_{x}+i\lambda\hat{\mathrm{e}}_{y})/\sqrt{1+\lambda^{2}}$ is the unit vector representing the laser polarization, with $\lambda=0$ for linear polarization and $\lambda=\pm 1$ for right- and left-handed polarizations.

It is convenient to rewrite the noninteracting part of the Kane-Mele-Hubbard Hamiltonian in Eq. (1) as an effective hopping term $\hat{T}_{0}=\hat{\mathcal{H}}_{\text{K}}+\hat{\mathcal{H}}_{\text{SOI}}=-\sum_{i,j,\tau,\tau^{\prime}}t_{ij}^{\tau\tau^{\prime}}\hat{c}_{i\tau}^{\dagger}\hat{c}_{j\tau^{\prime}}$, where the hopping amplitude is $t_{ij}^{\tau\tau^{\prime}}=\delta_{\tau,\tau^{\prime}}t_{1}$ for $i$ and $j$ that satisfy the NN condition and $t_{ij}^{\tau\tau^{\prime}}=\delta_{\tau,\tau^{\prime}}t_{2}-i\Delta\nu_{ij}\sigma^{z}_{\tau,\tau^{\prime}}$ for $i$ and $j$ that satisfy the NNN condition. With the Peierls substitution, the hopping part of the Hamiltonian gains an extra phase $t_{ij}^{\tau\tau^{\prime}}\rightarrow t_{ij}^{\tau\tau^{\prime}}e^{{i\frac{e}{\hbar}\bm{R}_{ij}\cdot\bm{A}(t)}}$, where $\bm{R}_{ij}=\bm{R}_{i}-\bm{R}_{j}$, $\bm{R}_{i}$ is the position of site $i$, $e$ is the charge of an electron, $\hbar$ is the reduced Planck constant and $\bm{A}$ is the vector potential of the laser pulse; $\bm{A}(t)=\frac{1}{2}(\bm{A}e^{-i\omega t}+\mathrm{c.c.})$, with $\bm{A}=\frac{iE_{0}}{\omega}\hat{\epsilon}$. The Peierls phase at time $t=0$ can be rewritten as $\frac{e}{\hbar}\bm{R}_{ij}\cdot\bm{A}\equiv\alpha_{ij}e^{i\theta_{ij}}$, with $\alpha_{ij}=\pm|\frac{e}{\hbar}\bm{R}_{ij}\cdot\bm{A}|$, such that $\alpha_{ij}=-\alpha_{ji}$, $\theta_{ij}=\theta_{ji}$, and $\theta_{ij}\in\left[0,\pi\right)$. Now, we can use the Jacobi-Anger expansion to rewrite the Peierls phase in the basis of its harmonics:

[TABLE]

where $\mathcal{J}_{m}(x)$ is an mth Bessel function of the first kind Kitamura et al. (2017).

In the presence of the laser field, the hopping term in the Hamiltonian depends on time, $\hat{H}(t)=\hat{T}(t)+U\hat{d}$. From Eq. (9), we find that $\hat{T}(t)=\sum_{m}\hat{T}_{m}e^{im\omega t}$, where $\hat{T}_{m}$ is the sum of all m-th Fourier modes of the hopping terms. We can additionally adopt the decomposition $\hat{T}_{m}=\hat{T}_{-1,m}+\hat{T}_{0,m}+\hat{T}_{1,m}$, where $\hat{T}_{dm}(t)$ changes the doublon number by adding $d$ double occupancies and is expressed as $\hat{T}_{dm}(t)=\sum_{n}\hat{P}_{n+d}\hat{T}_{m}(t)\hat{P}_{n}$. Since the hopping term is of second order in the creation and annihilation operators, it can change the double occupancy of the states only by $\pm 1$. Thus, we can express the hopping operator as

[TABLE]

To find the renormalized spin Hamiltonian in the strongly correlated regime, we first derive an effective static Hamiltonian using the Floquet formalism Maricq (1982); Rodriguez-Vega et al. (2018); Goldman and Dalibard (2014). To this end, we transform the original time-dependent Hamiltonian $\hat{H}(t)$ by using the canonical transformation $\hat{U}(t)=e^{-i\hat{S}(t)}$ Bukov et al. (2016); Kitamura et al. (2017),

[TABLE]

We can formally express $\hat{T}(t)=\eta\hat{T}(t)$, where $\eta$ plays the role of a bookkeeping parameter in the perturbation expansion. We expand $\hat{S}(t)=\sum_{\nu}\eta^{\nu}\hat{S}^{(\nu)}(t)$ and $\hat{H}^{\prime}(t)=\sum_{\nu}\eta^{\nu}\hat{H}^{\prime(\nu)}(t)$. We require the transformed Hamiltonian to be block diagonal in the doublon number operator $\hat{d}$. To fulfill this requirement, the unitary transformation $\hat{S}(t)$ must have the same periodicity as $\hat{T}(t)$; consequently, the transformed Hamiltonian $\hat{H}^{\prime}(t)$ will have the same periodicity as the original Hamiltonian $\hat{H}(t)$. Thus, we can write $\hat{S}^{(\nu)}(t)=\sum_{m}e^{im\omega t}\hat{S}^{(\nu)}_{m}$. With the further requirement that $\hat{S}(t)$ does not contain block-diagonal terms, we can uniquely determine the unitary transformation,

[TABLE]

where $\hat{S}^{(\nu)}_{d,m}$ changes the double occupancy number by $d$. We expand the transformed Hamiltonian of Eq. (11) into a power series in $\eta$ and determine $\hat{S}^{(\nu)}(t)$ iteratively in $\nu$ such that $\hat{H}^{\prime(\nu)}(t)$ is diagonal in the doublon number. After tedious but straightforward calculations, we obtain the transformed Hamiltonian up to the second order in the hopping parameter, $\hat{H}^{\prime}(t)=\hat{T}^{\prime}(t)+U\hat{d}$, where

[TABLE]

Now, we calculate the effective static Hamiltonian by time averaging the transformed Hamiltonian $\hat{H}_{\text{eff}}=\hat{P}_{0}\hat{H}^{\prime}(t)\hat{P}_{0}$, where $\hat{P}_{0}$ is the Gutzwiller projection onto the subspace containing no doubly occupied sites at all, i.e., the $d=0$ subspace Fazekas (1999). After some algebra, the effective static Hamiltonian is obtained in terms of the creation and annihilation operators:

[TABLE]

Note that to obtain this result, we have only considered the strongly correlated regime of the Kane-Mele-Hubbard model, $t_{1(2)}/U\ll 1$, and no assumption has been made on the range of $\omega$ and $\alpha_{ij}$. Using the relations in Eq. (6), we finally obtain the spin Hamiltonian at half filling,

[TABLE]

with the following renormalized spin-spin interactions,

[TABLE]

All spin interaction parameters are renormalized in the presence of a periodic drive by the same function, but $\alpha_{ij}$ differs between the NN and NNN parameters. Thus, the ratios between the renormalized NN and NNN parameters are different from those for the unperturbed parameters. Therefore, in a honeycomb lattice described by the Kane-Mele-Hubbard model, the ratio between the AFM exchange interaction and the intrinsic DMI changes during the light irradiation, while in a square lattice with the NN Rashba SOI, it is not possible to control this ratio. The renormalized spin interactions presented in Eq. (16) do not depend on the helicity in this model. In our model, perturbing the system with an AC electric field renormalizes the original DMI, which is already present in the unperturbed Hamiltonian. In the absence of a SOI, the unperturbed system does not display DMI and adding the electric field would not induce any DMI. This is different from the case studied in Ref. Bukov et al. (2016), in which it has been shown that an out-of-plane AC electric field, equivalent to a periodic time-dependent chemical potential, induces a DMI-like term in the system even in the absence of any SOI.

Figure 2 shows the dependence of the dimensionless NN exchange interaction $\tilde{J}_{1,ij}/J_{1,ij}$ and the dimensionless NNN DMI $\tilde{D}_{ij}/D_{ij}$ on the Floquet parameter $\mathcal{E}=\frac{eaE_{0}}{\hbar\omega}$, where $a$ is the lattice constant. We show this dependence for two different laser pulse frequencies. We set $\hbar=t_{1}=1$ and measure the energy in units of $t_{1}$ and the frequency in units of $t_{1}/\hbar$. The presented results correspond to the material parameters $t_{2}=0.1$ and $U=10$. Figure 2 shows that it is possible to not only change the sign and amplitude of the exchange interaction, as reported in Ref. Mentink et al. (2015), but also change the sign and amplitude of the intrinsic DMI. Figure 2 shows that the ratio $\tilde{D}_{ij}/\tilde{J}_{1,ij}\neq D_{ij}/J_{1,ij}$, which is responsible for ultrafast photoinduced spin dynamics phenomena, can be tuned by means of laser excitations in systems with specific symmetries. Here, we should emphasize that since in our model the NNN exchange interaction, the anisotropic exchange interaction, and the intrinsic DMI arise from NNN couplings, they are renormalized in the same way [see Eqs. 16(b)-16(d)].

To estimate the change in the ratio between spin-spin interactions $D/J_{1}$ within the current technology, let us consider the typical experimental parameters $U\approx 3$ eV, $t_{1}\approx 0.5$ eV, $\hbar\omega\approx 0.85U=2.55$ eV, and $a=4$ Å, and consider an electric field amplitude of $E_{0}=10^{9}$ V/m. Using these parameters, the DMI and the exchange interaction are renormalized as $\tilde{D}/D=1.047$ and $\tilde{J}_{1}/J_{1}=1.016$, respectively. Thus, the ratio between these two spin-spin interactions is also renormalized as $\tilde{D}/\tilde{J}_{1}=1.031(D/J_{1})$. These values are detectable experimentally. In Ref. Mikhaylovskiy et al. (2015), a change of $0.01\%$ in the ratio between the DMI and the exchange interaction has been reported by measuring the photoexcitation of the quasiantiferromagnetic mode in FeBO3.

Equations (Ultrafast control of spin interactions in honeycomb antiferromagnetic insulators) and (16) explicitly show that the spin Hamiltonian in the presence of a time-dependent external field can be effectively written as $\tilde{H}_{\text{S}}=H_{\text{S}}+g_{\alpha\beta ij}S^{\alpha}_{i}S^{\beta}_{j}E^{\alpha}E^{*\beta}$, where $\alpha$ and $\beta$ represent the spatial components of vectors, $i$ and $j$ refer to lattice sites, and $g$ is the optomagnetic coupling tensor, which can be read off from Eq. (16). Thus, the dielectric permittivity tensor, which determines the optical properties of the medium, is given by $\varepsilon_{\alpha\beta}=\partial^{2}\tilde{H}_{\text{S}}/\partial E^{\alpha}\partial E^{*\beta}$. The optomagnetic effect, which is described by the dielectric permittivity $\varepsilon$, can be detected by measuring the intensity of the light scattered by magnons, $I_{\mathrm{sc}}\propto(\varepsilon_{\alpha\beta}E_{0})^{2}$ O. Demokritov et al. (1985).

In ultrafast spin dynamics experiments, very intense laser pulses are used, and thus, it might be relevant to consider how heating might affect the validity of our approach. Recent theoretical Kuwahara et al. (2016); Kandelaki and Rudner (2018); Seetharam et al. (2018) and experimental Desbuquois et al. (2017) works have shown that the energy absorption rate is exponentially suppressed for high-frequency laser pulses, i.e., for $\hbar\omega/W\gg 1$, where $W\propto t_{1}$ is the fermionic bandwidth, and this condition holds in ultrafast experiments with optical laser pulses. Thus, rapidly driven systems have a very long prethermalization period, implying that the evolution of these systems in the presence of short laser pulses can be safely described by our formalism.

In summary, we have investigated the effect of intense high-frequency polarized laser pulses on 2D AFM spin-orbit Mott insulators using the Floquet theory. We have found that both the sign and the amplitude of the ratio between the DMI and the exchange interaction in a honeycomb lattice can be modified, regardless of the helicity of the laser pulse. In general, we have shown that this ratio might be renormalized only in systems with special lattice symmetries in which the DMI and Heisenberg exchange interaction in the spin Hamiltonian are originated from hopping integrals between different distance sites in the electronic Hamiltonian. Our calculations propose another way to achieve the ultrafast and energy-efficient control of spin-spin interactions and thus the engineering of topological objects and the topological properties of 2D van der Waals magnetic materials. The possibility of the ultrafast optical modification of the exchange interaction in bulk iron oxides has recently been reported Mikhaylovskiy et al. (2015). We hope that our work will motivate different ultrafast experiments on measuring both the exchange interaction and the DMI in 2D magnetic systems.

Acknowledgments. The research leading to these results was supported by the European Research Council via Advanced Grant No. 669442, “Insulatronics,” and by the Research Council of Norway through its Centres of Excellence funding scheme, Project No. 262633, “QuSpin.”