Unifying semiclassics and quantum perturbation theory at nonlinear order

We are thankful for the referee's thorough and comprehensive review of our manuscript. We are obliged for their positive comments about our work, notably,

The main strength of the paper is its clarity: the motivation is very clear and interesting to everyone. The motivation is conveyed in a very good fashion in the introduction. I find Preliminaries section very useful for the understanding of very technical derivation part. Even though the main focus of the paper is the formalism and its derivation, providing numerical examples improve the quality of the paper.

We are glad to implement all the changes requested by the referee.

1- An extended discussion on $\alpha$ is needed.

We are thankful to the referee for raising this point and we have now added a subsection in App. A (Fixing $\alpha$) that specifically deals with the matter of fixing $\alpha$. Without repeating our response to Ref. 1 on the same point, we restate here the main message regarding the expected value of $\alpha$. As $\alpha$ is a parameter which controls the degree by which nonlinear transport remains adiabatic, we imagine a limit scenario in which there are two quasiparticle bands, each with a lifetime $\tau_v$, $\tau_c$ representing the valence and conduction states, respectively. Following Phys. Rev. Lett. 125, 227401 (2020) we find that $\alpha = \frac{\frac{1}{\tau_v}}{\frac{1}{2\tau_v}+\frac{1}{2\tau_c}}$. The semiclassical limit is constructed as follows: we require infinitely sharp (clean) quasiparticle excited states, with $\tau_c \to \infty$ and $\tau_v \to \infty$. In order for the transitions to be delta-like, we also impose the limit $\tau_c / \tau_v \to \infty$, which also suggests a finite intraband relaxation as compatible with the semiclassical approach. However, this limit also imposes the condition that $\alpha \to 2$. We therefore conclude the semiclassical limit is \emph{only} obtainable from the perturbative formalism when $\alpha = 2$. Additionally we showed that the onset of semiclassics below the optical gap is also strictly dependent on $\alpha$, and can be used to extract it if necessary. In the simple tight-binding model that we used in this work, for practical evaluation of our terms, all broadening terms are naturally phenomenological. In a system where interactions, finite temperature, and disorder are taken into account, $\alpha$ may be determined directly from first principles, by evaluating valence and excited state lifetimes.

2- Authors use x instead of $\alpha$ in Figures 3 and 4. I think changing them to $\alpha$ would increase the readability.

We thank the referee for this observation. We apologize for the confusion. This is a typo that appeared during the generation of the figures. We have changed $x$ to $\alpha$ exactly as the referee suggests.

3 In the caption of Figure 5 and in the text below the Figure...

We thank the referee for this comment, and indeed we understand that this point was not made sufficiently clear in the main text. To remedy this, we have added a section to the Appendix that specifically deals with the nature of this local particle-hole symmetry. We recap the main points of this new section: in typical tight-binding models used for, there exists a form of k-local particle-hole symmetry, that acts as a reflection about the $E = 0$ axis. In simple systems this is represented by the operation $U C$ where $U$ is a model-dependent unitary and $C$ is complex conjugation. If two bands $n, n'$ are connected by this symmetry, then the reflection we alluded to has $\varepsilon_n(\mathbf{k}) = - \varepsilon_{n'}(\mathbf{k})$. Consequently, all $\sigma^{ab;c}$ in the semiclassical limit which depend on odd powers of the dispersion (or derivatives thereof, e.g. velocities) are odd under this symmetry: $\sigma^{ab;c}{\tau^2}$,$\sigma^{ab;c}$,$ \ldots$. Terms of odd power in $\tau$, such as $\sigma^{ab;c}{\tau}, \sigma^{ab;c}}$ are instead even. In the appendix, we provide a simple model calculation that shows this property.

4 - Correcting some typos:

We are obliged to the referee for catching these typos. We have since fixed them and with the help of Ref. 1's comments we have also corrected other mistakes and combed the manuscript for more errors of this type. 0

5 - The inline equation in page 4

We are agree entirely with the referee's suggestion. We have converted the inline equation into a full equation.

We are once again grateful to the referee for their insightful, serious and professional review of our manuscript.

We are grateful for the referee's report and especially thank them for their thorough combing of our manuscript. We would like to apologize for the number of typos and other typographic errors, which we have corrected with the referee's help. We conducted another search for errors and fixed a few additional typos. We would like to thank the referee for observing that our work provides

New results for second-order electrical conductivity and reproduction of previous results

and that our text is

Clearly written with sufficient description of technical details, and careful discussion of approximations and limitations of the theoretical framework with respect to previous works.

The referee asked:

My only question about this manuscript is about alpha which seems to be a phenomenological parameter. Is this the case? In principle, is it possible to determine the value of alpha for a particular model?

We answer: We thank the referee for raising this point and we understand that it was not made sufficiently clear in the original version of the text. To remedy this, we have now added an appendix (marked App. A in the new version) focusing on the role of $\alpha$ and how to fix it in the semiclassical limit. We now restate the main conclusions that we expand on in greater detail in the revised version of the text. As $\alpha$ encodes the information from interband and intra-band processes (Phys. Rev. Lett. 125, 227401 2020), it is a measure of the adiabaticity of transport in the presence nonlinear processes. In other words, let us consider a simple "two-band" limit, and assign quasiparticle $\tau_v$, $\tau_c$ relaxation times to the valence and conduction bands, respectively. Then the ratio $\alpha$ may be described by Matthiessen's rule, \begin{align} \alpha = \frac{\frac{1}{\tau_v}}{\frac{1}{2\tau_v}+\frac{1}{2\tau_c}}. \end{align} Clearly, in the limit $\frac{\tau_c}{\tau_v} \gg 1$ one obtains $\alpha \to 2$. Formally, this limit corresponds to the case of an infinitely sharp quasiparticle excited state, or an infinitely clean band in the single particle picture. It is precisely such a limit which gives the semiclassical formalism, since semiclassical equations of motion require well-defined bands, and sharp quasiparticle excited states ($\tau_v \to \infty, \tau_c \to \infty$, and $\frac{\tau_c}{\tau_v} \to \infty$). Therefore, the application of our results in the strict semiclassical limit is only possible when $\alpha = 2$. We have also looked at finite frequency expressions. There, the distance from the gap to the onset of semiclassics is controlled by $\alpha$ (Fig. 3., inset). In the simple tight-binding model we consider here for practical calculations, all broadening parameters ($\tau_v, \tau_c$ ) are indeed phenomenological. In models with realistic interactions, finite temperature and disorder it is possible to determine $\alpha$ in self-consistent manner.

We have looked up over and corrected all minor errors pointed out by the referee. We highlight a few examples below that the referee also phrased as questions. We stress again that we have fixed all the typos found by the referee.

p2, $\omega_{1,2}$ is not defined (perhaps I missed it?)

The two frequencies $\omega_1, \omega_2$ are explicitly introduced in Sec. II (A), as being related to the two electric fields perturbing the system, $E_a (\omega_1)E_b (\omega_2)$. Later we set the stage for either the dc component or the 2nd harmonic component $\omega_{1} = \pm \omega_{2} = \omega$. To prevent any ambiguity, we now add a sentence at the top-right side of page 2, which says

Here, $\omega_{1,2}$ refers interchangeably to the two frequencies of the electric field at 2nd order.

Note that "abelian" has / has not been capitalised in different parts of the manuscript.

We thank the referee for this observation and apologize for the inconsistency. Based on the nomenclature we found in the literature, we now prefer to keep "abelian" non-capitalized throughout.

p8, first column below Fig.2, too many brackets in $\sigma_{res}$ and $\sigma_{off-rest};$ also "off-rest" is probably a typo?

We thank the referee for this observation. This is indeed a typo which we have fixed to read "off-res".

p9, second column, ref to Fig.1 in final sentence --- should this be Fig.4?

We are obliged to the referee for spotting this important typo. Yes, we now correctly refer to Fig. 4.

p10, first sentence "As a natural extension of the procedure we offer Sec. III,..." ?

We have reformulated the introductory sentence of Sec. V, and removed any confusion regarding its meaning.

We would like to once again thank the referee for their in-depth and thorough review of our manuscript. The changes suggested have considerably improved our content and presentation.