Entropy Driven Inductive Response of Topological Insulators

We thank the referee for their comments and overall positive evaluation. Here, we give our point by point response to address referee's comments and questions:

1) "Why is the resulting interaction expected to be of the Heisenberg type, and not, e.g., dominated by Ising terms, depending on the detailed orbital angular momentum of the states? Is the behavior of all materials investigated in Sec. 3.2 expected to be the same in this respect? Etc."

• We thank the referee for their comment. In this manuscript, we focus on the Fermi contact interaction, which is the dominant interaction for electrons in s-bands. The orbital structure of topological surface states typically consists of a mixture of s- and p-type orbitals, and their exact relative weight varies across different materials. For electrons in p-bands, as the referee points out, the hyperfine interaction can indeed be more complicated. However, the energy scale of the hyperfine interaction for electrons in p-type orbitals is at least an order of magnitude smaller than that for electrons in s-type orbitals. Hence even if the bands mix, as long as the s-orbital component is significant, the dominant interaction will be the Fermi contact interaction. This was noted in a previous study that focused on the effective hyperfine interaction for the helical edge states in HgTe, which are a mixture of p-type and s-type orbitals, see. In the present work, for the sake of simplicity, we ignore the subleading contribution and assume that the electron/nuclear spin dynamics is dominated by the the contribution of the s-type orbitals. The effective interaction strength is determined by the weight of the s-band in the surface state orbital structure, multiplied by their bare interaction strength. This approximation is valid for materials with significant s-type orbital character, even if it is smaller than the p-orbital weight. To make our point more accessible to the reader, we have added the reference we mentioned above and two more references.

2) "..In reality, such entropy gain will also be achieved via nuclear spin diffusion into the bulk (through nuclear spin-spin interactions). Typically, this is a relatively slow process, but since the effect on the charge current found here is also very small it would be good to compare the relevant time scales more quantitatively."

* The referee is right in pointing out that spin-flip interactions via hyperfine coupling is not the only mechanism causing a change in nuclear polarization: dipole-dipole interaction between nuclear spins leads to a diffusion of the nuclear spin (polarized by the topological surface state) into the bulk of the material. This mechanism changes the entropy of the nuclear spin subsystem as well. However, the energy scale of the dipole-dipole interaction is orders of magnitude lower than the Fermi contact interaction. As an example, we consider two Bismuth nuclear spins ($\gamma_{Bi}$: 6.9 MHz/T, [see Ref.](https://doi.org/10.61092/iaea.yjpc-cns6), spaced apart by the lattice constant of BST $a_0 =$ 0.439 nm. The resulting dipole-dipole interaction [given in Ref.](https://doi.org/10.1016/S0167-6881(98)80007-4) is proportional to $\mu_0 \gamma_1 \gamma_2 \hbar^2/(4\pi r^3) =$ 8.310$^{-14}$ eV. In comparison, the Fermi contact interaction is proportional to $\lambda = A_0 v_0/\xi =$ 2.5 $\mu$eV nm$^2$, which is multiple orders of magnitude larger than the dipole-dipole interaction. Furthermore, we point out that the orders of magnitude difference between Fermi contact interaction and dipole-dipole interaction is quite generic. Therefore we assume that entropy loss via diffusion through dipole-dipole interaction is negligible on the timescales of the inductive effect.

3) "Since I consider the pedagogic style of the manuscript one of its key values, I would recommend adding one initial step at the beginning of App. B, defining the greater and lesser Green functions and showing how (67,68) follow. This would balance that part with the very detailed introduction provided around Eqs. (3-7)."

• We thank the referee for their comment. Following their suggestion, we have extended the discussion around Eq. 67 and 68, showing how to get the lesser and greater Green's functions and calculate them.

* We added two more references about the effect of s- and p-orbital mixing on the hyperfine coupling in Section 3.2. We also made Ref. 26 accessible.
* Following referee's suggestion, we extended the discussion about the in Appendix B.
* We fixed a couple of typos.