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

1. We thank the Referee for this comment, and in the following motivate our choice of model. In the revised manuscript, we focus on the leading-order light-induced phenomena in generic honeycomb ferromagnets. Both next-nearest neighbor and spin-orbit interactions are straightforward to add, but the former will not affect the topology of the magnon bands. Since our system is inversion symmetric this also rules out most spin-orbit interactions, the Kane-Mele interaction being an important exception. To our knowledge there are no estimates of the magnitude of such an interaction in magnetic vdW materials, and we therefore neglect it as a first approximation. Under the assumption that spin-orbit interactions are negligible, the equilibrium scalar spin chirality interaction will vanish since it is proportional to the imaginary part of $t_1^2t_2/U^2$ [Kitamura, Oka and Aoki, Phys. Rev. B 96, 014406 (2017)]. We would like to note that our results hold irrespective of additional terms being present or not. If additional terms are present, and compete with our scalar spin chirality, the respective order of magnitude of these terms can be compared with the one in our manuscript to see which one dominates. Thus, although we agree with the Referee that both spin-orbit interactions and next-nearest neighbor hopping could be of importance in real vdW materials, we believe our work provides an important first step to describe light-induced phenomena in generic honeycomb ferromagnets.

2. We thank the Referee for spotting this difference, and hope the following explanation is satisfactory: The reason why a DM interaction is found in the work of Bukov, Kolodrubetz and Polkovnikov (BKP) and not in the present work, is due to the different symmetry breakings induced by the driving schemes. In the present case, we consider a circularly polarized laser field coupling to the electron kinetic term via Peierls substitution, which preserves the SU$(2)$ spin symmetry of the system and leads to isotropic spin parameters. In contrast, the BKP paper considers a local Zeeman coupling to the $z$-component of the spin, and employs a field that is different along the $x$- and $y$-directions of the lattice. This term explicitly breaks both the SU$(2)$ spin symmetry and the $C_4$ rotational symmetry of the system. We note that a similar DM interaction as in the BKP paper can be induced by a spin-optical coupling based on the Aharanov-Casher effect [Owerre, Journal of Physics Communications 1(2), 021002 (2017)], where the direction of the DM vector is determined by the anisotropic coupling to the spin operators. In the present case, where the light-matter coupling is isotropic, the DM interaction is replaced by the SU$(2)$ invariant scalar spin chirality term.

3. We agree with the Referee that biquadratic (and other) terms could be of importance in CrI$_3$, as argued in several recent studies [Chen et al., Phys. Rev. X 8, 041028 (2018), Lee et al., Phys. Rev. Lett. 124, 017201 (2020), Kartsev et al., arXiv:2006.04891]. We have therefore followed the Referee's suggestion and focused the revised manuscript on generic honeycomb ferromagnets. The discussion of CrI$_3$ as a possible material realization (pointing out some of the subtleties in extending the model to $S = 3/2$ systems) has been deferred to the section about possible experimental realizations.

4. Within our model there is no qualitative difference between the magnons edge states of zig-zag and armchair ribbons, in the sense that their topological properties are unaffected. However, just like in the electronic Haldane model, the ${\bf K}$ and ${\bf K}'$ points of the two-dimensional Brillouin zone are projected onto different momenta of the surface Brillouin zone for a zig-zag ribbon, while they correspond to the same surface momentum for an armchair ribbon. This could have important consequences when trying to populate the magnon states, since this process is typically restricted to small momentum transfers.

5. We thank the Referee for pointing this out, and have made sure the notation is unique and consistent in the revised manuscript.

6. We thank the Referee for pointing this out and have added a reference to the original work by Hubbard [Hubbard, Proc. R. Soc. Lond. A 281 401–419, (1964)].

7. We thank the referee for pointing out these reference, to which we have compared our results and included those we found particularly relevant.

1. We agree with the Referee that the model considered in the manuscript is too simplistic to describe the detailed magnetic properties of CrI$_3$. However, we believe it still bears merits as a starting point for more detailed studies of light-induced magnetic phenomena in magnetic vdW materials. We have therefore, in the revised manuscript, shifted the main focus to light-induced topological magnons in generic $S = 1/2$ honeycomb ferromagnets, and included a discussion of CrI$_3$ as a possible material realization (pointing out some of the subtleties in extending the model to $S = 3/2$ systems).

2. We agree with the Referee that there is a strong analogy between the properties of the electronic and bosonic Haldane models, and how they may be realized out of equilibrium. In the revised manuscript, we have included a citation to the original work by Oka and Aoki.

3. Although we agree with the Referee that the light-matter coupling considered in the manuscript is of a different physical origin than a coupling based on the Aharanov-Casher effect, we believe a direct comparison between the two is both possible and well defined. Therefore, we here elaborate on how this comparison is made and what it implies. In both approaches, driving the system leads to additional phase factors for interactions along on a bond, whose strength can be quantified by a dimensionless coupling constant $\lambda$. Moreover, the value of $\lambda$ is what controls the photo-induced renormalization of the spin parameters, as it appears as the argument of the Bessel functions defining the non-equilibrium spin parameters. In the manuscript we show that our light-matter coupling leads to a value $\lambda_e = (eEa)/(\hbar\omega)$, while the corresponding value arising from an Aharanov-Casher coupling is $\lambda_m = (g\mu_B Ea)/(\hbar c^2)$ [Owerre, Journal of Physics Communications 1(2), 021002 (2017)]. It is then straightforward to verify that for optical wavelengths and a laser field strengths of $E \approx 10^9$ V/m, the ratio of the two couplings $\lambda_m/\lambda_e = (g\mu_B \omega)/(ec^2) \approx 10^{-5}$. The value of this ratio is not only of academic interest, but has important physical consequences in particular for the magnitude of the photo-induced topological gap. Since even for strong lasers fields we find $\lambda < 0.1$, the Bessel functions defining the spin parameters can be expanded to show that the magnon band gap $\Delta \sim \lambda^2$. Thus, compared to a driving scheme exploiting the Aharanov-Casher effect, the band gap we find is enhanced by a factor $10^{10}$. This enhancement will be of crucial importance for any experimental realization.

Minor points: The values shown in the spectral functions of Fig. 3 are normalized to the highest value, which is explained more clearly in the figure caption of the revised manuscript. We've also inserted a color bar for clarity.