Light-induced topological magnons in two-dimensional van der Waals magnets

Driving a two-dimensional Mott insulator with circularly polarized light breaks time-reversal and inversion symmetry, which induces an optically-tunable synthetic scalar spin chirality interaction in the effective low-energy spin Hamiltonian. Here, we show that this mechanism can stabilize topological magnon excitations in honeycomb ferromagnets such as CrI$_3$ and in optical lattices. We find that the irradiated quantum magnet is described by a Haldane model for magnons that hosts topologically-protected edge modes. We study the evolution of the magnon spectrum in the Floquet regime and via time propagation of the magnon Hamiltonian for a slowly varying pulse envelope. Compared to similar but conceptually distinct driving schemes based on the Aharanov-Casher effect, the dimensionless light-matter coupling parameter $\lambda = eEa/\hbar\omega$ at fixed electric field strength is enhanced by a factor $\sim 10^5$. This increase of the coupling parameter allows to induce a topological gap of the order of $\Delta \approx 2$ meV with realistic laser pulses, bringing an experimental realization of light-induced topological magnon edge states within reach.

In addition to the intrinsic topological properties of vdW magnets, the tremendous progress in functionalization of materials through light-matter coupling [14][15][16][17][18][19] shows that it is possible to manipulate the magnetic and topological order of such materials using laser fields. In recent theoretical studies it has been shown that driving a two-dimensional Mott insulator with circularly polarized light breaks both time-reversal and inversion symmetries. This is reflected by an induced scalar spin chirality interaction that governs the transient dynamics of low-energy spin excitations [11,20]. Remarkably, optical irradiation red-detuned from the Mott gap can limit heating and absorption to enable a controlled realization of such Floquet-engineered spin dynamics, and it has been argued for a Kagomé lattice antiferromagnet that the spin chirality term leads to a chiral spin liquid ground state in herbertsmithite and kapellasite [11]. Experimental realizations of Floquet-engineered spin Hamiltonians have also been demonstrated for both classical [21] and quan-tum magnetism [22] using ultracold atoms in driven optical lattices [23].
In this work, we demonstrate that the photo-induced scalar spin chirality has consequences for the low-energy magnetic excitations of ferromagnetic systems. In particular, for honeycomb ferromagnets such as monolayer CrI 3 , it leads to a magnon Haldane model [24] with a topological gap and chiral magnon edge states [25]. To this end, we first derive the magnitude of the induced time-reversal symmetry breaking contribution for a honeycomb Mott insulator. We then show that application of the effective spin Hamiltonian to the prototypical monolayer vdW magnet CrI 3 [2,[26][27][28] can lead to a gap ∆ ≈ 2 meV in the magnon spectrum for a realistic field strength E = 10 9 V/m and photon energy ω = 1 eV, inducing non-zero Chern numbers and leading to chiral magnon edge states. Importantly, we find that the dimensionless Floquet parameter that describes the magnitude of light-matter interaction is enhanced by a factor ∼ 10 5 compared to similar but conceptually distinct driving schemes based on the Aharanov-Casher effect for pure spin models [25,29], since the electric field couples to the charge instead of the magnetic moment. This amplification is shown to be crucial for a potential experimental realization of a topological magnon phase in monolayer vdW magnets.
Model.-To assess the magnitude of photo-induced time-reversal symmetry breaking for honeycomb Mott insulators, we commence by deriving an effective transient spin-1/2 Hamiltonian from a single-band Mott insulator where c † iσ creates an electron at site i with spin projection σ, t is the hopping amplitude between nearest neighbor sites i and j, and U 0 is a local interaction. We also consider nearest neighbor direct and exchange interactions V and J D , the later being expressed in terms of the spin operatorŜ i = c † iσ τ σσ c iσ [30] where τ is the vector of Pauli matrices. We use the Einstein summation convention for repeated spin indexes.
The electrons interact with an external electromagnetic field described via the Peierls phases To break time-reversal symmetry and induce a scalar spin chirality, we use a circularly polarized laser in the dipole approximation with vector potential ∂ t A(r, t) = −E(t)(cos ωt, ζ sin ωt), where ζ = ±1 for right/lefthanded polarization. Assuming a constant envelope E(t) = E and writing δ ij = r i − r j = a(cos φ ij , sin φ ij ) with a the lattice constant, the Peierls phases are θ ij (t) = −λ sin(ωt − ζφ ij ). The dimensionless quantity λ = eEa/ ω determines the effective field strength of the laser. In an optical lattice, eE is replaced by the driving force F , which may result from an acceleration of the lattice [31] or a magnetic field gradient [32]. Although the above model provides a simplified description of realistic monolayer vdW magnets, neglecting both the multi-orbital structure of the transition metals ions and the superexchange processes induced by interactions with the surrounding halides [33], it provides a starting point for more advanced treatments. Further, since the topological properties of honeycomb ferromagnets are determined by the lattice structure and the presence or absence of time-reversal symmetry [24], we expect the model to give a correct description of the topological features of the magnon excitations.
Effective spin Hamiltonian.-We now construct an effective spin Hamiltonian for driving frequencies J ω U , where J ∼ t 2 /U is the leading order Heisenberg exchange in equilibrium. We have followed the method of Ref. [11] to obtain the effective Hamiltonian to fourth order in t/U for a periodic external field. For a slowly varying envelope E(t) the Hamiltonian is almost periodic with the period H(t + 2π/ω) = H(t). This allows us to employ Floquet theory and rewrite the electronic Hamiltonian exactly using a Fourier expansion expressed in the product space of the electronic Hamiltonian and the space of periodic functions [34] denoted by Fourier modes |m , which can be identified with the classical limit of m absorbed or emitted virtual photons. Here, J m (x) is the Bessel function of the first kind of order m. The interaction Hamiltonian is where the nearest neighbor direct interaction has been absorbed by a renormalization of the Hubbard U [35]. Using quasi-degenerate perturbation theory to simultaneously integrate out the doubly occupied states and the m = 0 Floquet states [11,36], the effective honeycomb lattice spin Hamiltonian corresponding to the electronic system is given to fourth order in t/U by Here J and J are respectively the nearest and nextnearest neighbor light-induced Heisenberg exchanges, and χ is a synthetic scalar spin chirality. A non-zero value of χ signals a non-coplanar spin texture and can appear in equilibrium due to e.g. Dzyaloshinskii-Moriya interactions or geometric frustration [37,38]. For electrons hopping around closed loops in such a spin texture the spin chirality acts as an effective magnetic field that can give rise to the topological Hall effect [39]. The full expressions for the spin parameters are given in the Supplemental Material (SM). We note that J has contributions from all even orders in t/U , while J and χ appear only at fourth order. On the honeycomb lattice, a non-zero spin chirality arises due hopping processes that enclose an isosceles triangle within the hexagons, as indicated schematically in Fig. 1a (and discussed further in the SM). Such processes lead to a net phase accumulation in analogy with electrons moving in closed loops in an external magnetic field, and lead to time-reversal symmetry breaking. However, in contrast to using an external magnetic field, driving with a circularly polarized electric field conserves the SU (2) spin symmetry. In the non-interacting limit the corresponding complex next-nearest neighbor tunneling has already been implemented in optical lattices using circular driving [40].
Justifying the Hamiltonian for CrI 3 and optical lattices.-Below we use the spin Hamiltonian to study the magnon excitations of driven monolayer CrI 3 . Since our effective Hamiltonian was derived for S = 1/2, it gives a simplified description of S = 3/2 ferromagnets such as CrI 3 . However, similar spin Hamiltonians have been used to successfully describe the magnon excitations in CrI 3 [26,41].
The main effects of including the t 2g orbitals of Cr 3+ via a Kanamori-Hubbard model (except for an obvious renormalization of the spin parameters), are spin-orbit coupling and the appearance of biquadratic exchange terms (Ŝ i ·Ŝ j ) 2 [42,43]. Biquadratic exchange can generate nematic instabilities [44,45] and break the C 6 rotation symmetry down to the C 3 subgroup, generating a trivial mass term that competes with Haldane mass generated from the breaking of time-reversal symmetry and can trivialize the magnon band topology. To estimate this effect, we performed density functional theory calculations in the DFT+U formalism with the Octopus code [46,47] to estimate the size of the trivial mass term in monolayer CrI 3 (with the value of U self-consistently determined via the ACBN0 hybrid functional [48]). We find a ground state with C 6 symmetry to a numerical accuracy 10 −6 , and thus conclude that the effects of the biquadratic terms are negligible in CrI 3 .
Ultracold fermions in optical lattices naturally realize the Hubbard Hamiltonian, and V and J D are typically negligible [49]. Nevertheless, ferromagnetic spin models can be implemented using near-resonant periodic driving [22]. For most systems, S = 1/2 (where we expect our results to still hold approximately), but magnetic correlations for larger S have also been observed using alkali-earth-like atoms [50].
Antiferromagnetic systems.-In the following we assume |χ| |J|, so that depending on the sign of J the system is either ferromagnetic (J < 0) or antiferromagnetic (J > 0). It has previously been shown that topological magnon edge states can be induced by a constant electric field gradient that splits the magnon bands into Landau levels and leads to a magnon version of the quantum (spin) Hall effect in (anti-) ferromagnets [51,52]. In the present work the homogeneous but time-dependent electric field instead opens a gap at magnon band crossings, leading to a magnon analog of the quantum anomalous Hall effect. In the antiferromagnetic regime we find that the bands are nearly degenerate with no crossings, and the system remains in a topologically trivial phase. This agrees with previous work where the edge modes of the Néel state were shown to be topologically trivial [53]. However, by adding an in-plane magnetic field [53] or considering an antiferromagnetically coupled bilayer [54], topological magnon edge states can be induced. Although we focus below on the ferromagnetic state, we expect that an application of our formalism to the noncollinear and bilayer antiferromagnetic cases would lead to similar conclusions.
Magnons on the honeycomb lattice.-We denote the lattice vectors of the honeycomb lattice by b i and the vectors between nearest neighbor sites by δ i (see Fig. 1b). On the honeycomb lattice the angles Φ ik = ζ(φ ij − φ kj ) between next-nearest neighbor sites are given by Φ ik = 2πζν ik /3, where ν ik = 1 (ν ik = −1) for hopping in a clockwise (anti-clockwise) direction (see Fig. 1b). This leads to a spin chirality of the form χ ik = ζν ik χ where the sign alternates depending on the bond direction.
For a ferromagnetic ground state we can solve the system to leading order in S −1 using the Holstein-Primakoff 2Sa i for the spins on sublattice A, and similarly but with a i → b i for the spins on sublattice B. In terms of its Fourier components the Hamiltonian can be writ- h y , h z ) and τ is the vector of Pauli matrices. The eigenvalues of this matrix are ± (k) = h 0 (k) ± h(k) · h(k), where h 0 = 3JS + 6J S + 2J Sξ k , h x + ih y = −JSρ k and h z = 2ζχSσ k . To simplify the notation we have defined the quantities ρ k = i e −ik·δi , ξ k = i cos(k · b i ) and σ k = i sin(k · b i ). In Fig. 1c we show the magnon band structure for equilibrium spin parameters of bulk CrI 3 [26]. For χ = 0 the system has Dirac points at K η = η(4π/3 √ 3, 0) (where η = ±), while for χ > 0 a gap is opened of magnitude ∆ = 6 √ 3χS. As shown below, the gap opening is associated with a transition to a non-trivial topological state.
Chern numbers and edge states.-To determine the topological structure of the system for χ > 0, we calculate the Chern numbers of the dressed magnon bands. Since the dominant contribution to the Berry curvature comes from the regions around the Dirac points, we expand the Hamiltonian around K η . To linear order in κ = k − K we find the Hamiltonian H η = vηκ x τ x −vκ y τ y +wητ z , where v = 3JS/2, w = −3 √ 3ζχS, and we have neglected constant terms proportional to J and J .
The Berry potential for the quasi-stationary state is obtained from the expression [55] v η s (k) = Im where |ψ ks are the eigenstates of H η . It is clear from the cross product that v −s = −v s and so it is sufficient to compute v + . We calculate the matrix elements by noting that ∂ κx H η = vητ x and ∂ κy H η = −vτ y , and defining d 2 = h·h = 1 4 ( + (k)− − (k)) 2 we find the Berry potential v η s = −sηv 2 h z /(2d 3 ). Since the Chern number is the integral over the Berry potential we have For positive χ the upper (lower) band has a Chern number C = ζ (C = −ζ).
The non-zero Chern numbers imply the existence of topological magnon edge states. We verify this explicitly for a ribbon geometry with zig-zag edges, periodic in the x-direction and with n y sites in the y-direction. In Fig. 1d we show the band structure of the ribbon for n y = 100, where chiral edge states are situated in the bulk band gap and connect the Dirac points at K + and K − .
We note that in contrast to the edge states of a quantum spin Hall insulator, the edge magnons of different chirality are located on opposite edges of the sample. Since the sign of the Chern numbers and thereby the chirality of the edge states are determined by the polarization of the optical field, this allows to control the propagation direction of the edge magnons by changing the helicity of the field.
Parameter dependence of the scalar spin chirality.-We have seen that a non-zero value of the scalar spin chirality χ leads to a topological magnon state. We now discuss the values of the frequency and electric field strength needed to induce this state in monolayer CrI 3 .
We start by considering the ratio χ/J as a function of the photon energy ω and effective field strength λ, which is a measure of the ratio between the bandgap and the bandwidth. The results are shown in Fig. 2a for the electronic parameters t = 50 meV, U = 1.25 eV and J D = 12 meV corresponding approximately to monolayer CrI 3 [56]. We find a resonant behavior in χ/J when the frequency ω = U/n with n integer, which corresponds to the thresholds for n-photon excitation across the Mott gap. In addition, the diagonal feature extending across Fig. 2a indicates the transition from a ferromagnetic to an antiferromagnet effective exchange parameter [36]. Approaching this transition while simultaneously ensuring |χ| > |J | will bring the system into a state dominated by the spin chirality term. This could potentially lead to new exotic physics such as a skyrmion lattice [57] or chiral spin liquid ground state [11,58].
Naively these results suggest employing a sub-gap driving protocol that exploits the resonant enhancement of χ for ω ≈ U/n while simultaneously minimizing electronic interband transitions. However, numerical studies have shown that for driving frequencies close to the multiphoton resonances the system heats immediately and the spin description becomes invalid [11]. In addition, since the real Mott gap is not at U but at the slightly smaller value U − xt (with x a numerical factor of order unity), the frequency has to be chosen below this gap to avoid heating. In the following we therefore focus on photon energies ω/U ≈ 0.8, which is below the Mott gap for x < 5.
Assuming a realistic field strength E ≈ 10 9 V/m, U = 1.25 eV, a = 5Åand ω ≈ 1 eV, an effective field strength λ ≈ 0.5 can be achieved. Because interband transitions are avoided in this driving protocol, larger field strengths may still yet be applied without inflicting material damage or other detrimental effects that would disrupt the induced scalar spin chirality. However, even for λ ≈ 0.5 it is possible to open a bandgap of magnitude ∆ ≈ 2 meV (see Fig. 2b). In contrast, a treatment based on the Aharanov-Casher effect in pure spin systems leads to a field strength λ m = (gµ B Ea)/( c 2 ) [25], which is smaller than the electronic equivalent by a factor λ m /λ e = (gµ B ω)/(ec 2 ) ≈ 10 −5 . Since χ ∼ λ 2 for small λ, this leads to a reduction of the gap size by about 10 −10 making an experimental realization of topological magnon systems based on the Aharanov-Casher effect highly challenging. In contrast, the driving protocol proposed here opens a topological gap well within reach of experimental probes.
In optical lattices, heating rates have been shown to be manageable even for λ > 1 [59]. The magnon band gap can hence be enhanced to values above the currently accessible temperature scales [60].
Validating the Floquet treatment.-To validate the Floquet treatment we compare the results obtained via the static Floquet Hamiltonian with numerical results from time-propagating the system with a quasi-periodic spin Hamiltonian (for details see SM). We take the external field to be switched on adiabatically over approximately 350 periods T = 2π/ω, after which we propagate the system for an additional 1750 periods.
To visualize the magnon edge states we consider the spectral function A k ( , t). For a non-equilibrium system the time-dependent spectral function is defined by The lesser Green's function is proportional to the distribution function f of the initial state, and therefore G < k (t, t ) = 0 since we start the time-evolution from the magnon ground state. The greater Green's function is given by where |sk(t) = U (t)|sk are the time-evolved singlemagnon eigenstates |sk of the equilibrium Hamiltonian, is the time-ordered evolution operator, and the trace is over all single-magnon states. Since G k is diagonal in k, we can calculate the spectral function by separately time-propagating the states |sk for each k. In equilibrium |sk(t) = e −i sk t |sk , and we find the Floquet spectral function In Fig. 3 we compare the Floquet spectral function A F k ( ) and the non-equilibrium spectral function A k ( , t) for a ribbon with n y = 20. We find a very good agreement between the Floquet and non-equilibrium spectral functions, indicating that for the given parameters the static Floquet Hamiltonian provides a good description of the non-equilibrium magnon dynamics.
Suggested experiments.-We end the paper with a discussion of possible experiments that would support the presence of a topological magnon band structure in a driven system. We note that so far, there has been no experiments that address topological magnons in a nonequilibrium setting. However, in equilibrium studies on ferromagnetic bulk CrI 3 and antiferromagnetic Cu 3 TeO 6 have found a magnon band structure consistent with a non-trivial topology attributed to either next-nearest neighbor Dzyaloshinskii-Moriya interactions [26], nearest neighbor Kitaev interactions [61], or the lattice structure [62].
As shown above, a non-zero scalar spin chirality leads to a topological gap, and can generally be probed by Faraday or Kerr rotation measurements [20]. The associated gap opening at the K-point in the magnon dispersion will affect the two-magnon optical excitation spectra, as probed by THz spectroscopy [63] or Raman and Brillouin spectroscopy [64]. However, the details of the optical spectra in these types of experiments would require dedicated calculations. The magnon edge states could potentially be probed directly using non-local magnon transport techniques, where magnons can be (detected) injected via the (inverse) spin Hall effect in platinum strips [65]. Finally, resonant inelastic X-ray scattering can be used to probe the magnon dispersion [66].
In optical lattices, a spectroscopic probe could be implemented using oscillating magnetic field gradients. In addition, static gradients can be used to imprint magnons with specific wavenumbers. Their subsequent dynamics gives access to the magnon dispersion and can be probed using spin-and site-resolved detection [67].
Conclusions.-To summarize, we have demonstrated that non-equilibrium driving based on periodic laser fields coupling to charge degrees of freedom can induce topological magnon edge states in the spin sector of prototypical two-dimensional quantum magnets. Specifically, for the recently discovered monolayer van der Waals magnet CrI 3 , we predict that a scalar spin chirality term can be induced leading to a sizeable magnon bandgap under realistic driving conditions. This opens the door for potential all-optical topological spintronics applications.
However, an important open problem for future studies is the question of how magnon edge states can be populated in a controlled fashion. Here we note that the situation is different compared to optical engineering of electronic systems, where the generation of dressed Floquet bands, their population, as well as the associated material heating, are intimately linked [18]. In the present work the separation between photon and magnon energy scales means that driving does not automatically populate the magnon bands, and as discussed above heating is largely avoided by adopting a sub-gap driving protocol. Populating the magnon states thus becomes a separate issue to be dealt with in addition to the generation of the non-trivial Floquet bands. We note that population by direct optical pumping have been discussed and experimentally verified for chiral edge states in topological exciton-polariton systems [68][69][70][71][72]. However, direct optical population of chiral magnon edge states through dipolar excitation is usually not possible and one should rather explore indirect mechanisms, for instance through two-magnon Raman scattering.
As an alternative route towards engineering topological magnon edge states with light, non-classical photon fields in cavities can be employed to control magnetic exchange interactions [73,74] and induce nontrivial topology with chiral light modes [75,76]. We also envisage the possibility to combine optical engineering with the control offered by bilayer Moiré systems [77] to induce and control topological magnons.