Bloch-Lorentz magnetoresistance oscillations in delafossites

We thank the referee for their report. The referee identified several problems with the presentation of the manuscript, most importantly its relation to the Kubo theory presented in the work of Putzke et al., as well as the benefits it adds. We have critically reassessed the manuscript and we agree with the reviewer that it could be significantly improved, which we did our best to do now. The most important changes in the manuscript are given in the list of changes.

Below we also provide the answers to the specific queries by the referee.

1) I find the motivation for the manuscript rather confusing. The authors claim that Putzke et al.’s theory “reproduces the results but offers limited insight into the nature of the phenomenon”. To be quite frank I think this statement is misleading. Putzke et al. explain clearly the underlying physics of their theory in terms of a diffraction-like grating effect of the hopping between adjacent layers, based on this picture they then perform theoretical calculations using the Kubo formula and it fits very well with their experimental findings (Fig. 4 of Ref. 12). Putzke et al. do admit they are open to alternative explanations at the end of their manuscript, however, the current authors should clearly explain: i) What additional mechanistic insights are gained from their semi-classical explanation? ii) What experimental observations can their calculation explain which the mechanism of Putzke et al. cannot? iii) Conversely, are there aspects of the experiment that Putzke et al.’s theory can better explain? iv) Do the two theoretical pictures connect with each other or are they contradictory?"

By definition, a Kubo approach constructed with the full Hamiltonian is exact and sufficient to explain the complete behaviour of this system. We, therefore, do not dispute the validity of the Kubo approach. On the other hand, the full Kubo approach would require simulating the full sample cross-section down to the wavelength resolution, which is far beyond modern computation capacity.

Additionally, the Kubo approach requires a derivation of the transition rates, which was not performed in the work of Putzke et al. Our approach identifies the Fermi wavelength as a parameter that is irrelevant to the phenomenon and demonstrates that this physics occurs in a purely classical system. We, therefore, identify an important simplification of the Kubo model. Naturally, if for example, quantum oscillations were the goal, our approach would not capture those, while the Kubo formalism would.

Because our theory is semiclassical, it proves that the oscillations do not rely on phase coherence beyond the lattice scales. This is compatible with the persistence of oscillations in an unusually high temperature observed by Putzke et al. The possible semiclassical origin of the oscillations was overlooked in the previous analysis. Specifically the abstract states: "These results demonstrate extraordinary single-particle quantum coherence lengths in the delafossites". We have also found out that similar magnetoresistance oscillations were observed in millimeter-sized samples of ballistic gallium at 1.3K in Ref 14. This serves as additional evidence against the coherent nature of the phenomenon.

Our result also bears physical interest because it is a rare example of a case where $\hbar$ appears without interfering paths. This happens because the quantum mechanical nature of the system is encoded in its periodic Fermi surface. Our approach also allows us to directly model both the sample geometry and the relative impact of different scattering mechanisms.

2) The authors admit that a ballistic mean free path e.g. the ~ 20 um found in Ref. [8] would rule out their mechanism since the oscillations suggested by the authors would become “fully visible” in such a case. The authors then propose a workaround which is scattering from the sample walls. I think the authors should explain the basis on which they claim this is the “dominant mechanism” for scattering. To be precise, do they believe there is any experimental data in Putzke et al. to support this assertion? Further, how do the authors reconcile their claim with the experimental observation (pg. 4 end - start pg. 5 of Ref. 12) that in samples where the mfp was reduced from 20 um to 1 um by irradiation oscillations were not visible in 8 um wide samples but became visible only when the sample was narrowed to 1 um (i.e. ballistic)? (If boundary scattering is dominant then why does a narrower sample exhibit the oscillatory behaviour but a wider one does not?)

Upon rewriting the manuscript we realized that this part of the analysis was not explained clearly. We summarize the argument below. The experiments of Putzke et al. were performed in an effectively ballistic sample with a small aspect ratio. Our model applies to this case and establishes that the aspect ratio explains the observed magnetoresistance. If the sample had a mean free path shorter than its width, as described in Ref. 12, then the bulk scattering would become the dominant mechanism determining the oscillation visibility. Therefore the situation described in the referee's query is in complete agreement with our analysis. We do not claim the contrary in the current version, nor did we in the previous submission.

3) The authors have a discussion on titled magnetic fields which concludes that the fact that oscillations disappear when the cyclotron orbits fit into the sample. I was surprised to see that at no point in section 3.3 do the authors mention that Putzke et al. also discuss this and indeed show very nice data of the transition from linear in B oscillations to SdH like oscillations when the field angle is tilted. The authors should outline why their discussion on this matter differs from Putzke et al. or, if the mechanism is the same (which is appears to me they are), they should acknowledge this fact

Section 3 of our manuscript compares the predictions of the semiclassical theory with the measurement reported in the work of Putzke et al. The authors have indeed identified the relevance of cyclotron radius in controlling oscillation appearance, which we now clearly acknowledge in the manuscript, and we apologise for not clearly stating this earlier. The relation between the two approaches is the same in this case as we have explained previously.

4) At no point is the reader shown what “full visibility” of the purported oscillations in conductance look like for a clean system, how the oscillations looks with decreasing mfp, or the difference in appearance of the oscillations with/without the dominant boundary scattering. All of these are key messages of the paper.

The full visibility oscillations are given by the Eq. 22 of the manuscript, which has $\sigma_{zz} \propto (1 - \cos(\omega B_y))$. We believe that this function is sufficiently simple to not require a plot. The previous version of the manuscript demonstrated the effect of the sample aspect ratio in Fig. 3, which we have extended with a similar plot for the mean free path in the resubmitted version.

5) The authors state: "The extension of the theory to a realistic sample geometry is straightforward... but it strongly increases the computational costs and therefore we consider it unjustified for the purposes of our study." I do not understand why the numerical integration of Eq. (29) becomes so costly for a realistic sample geometry and, since the whole message of the manuscript is that the the authors can example the experimental data, it seems strange to claim it is unjustified to show their calculation for the correct experimental geometry.

We have now explained that combining the full sample geometry with an out-of-plane magnetic field requires a 4D integral. While it is possible to carry it out, unlike a fully quantum-mechanical simulation on the scale of the Fermi wavelength, we believe that the added benefit is minimal, and therefore model this approximately. This approximation is qualitatively appropriate, as demonstrated by Figure 3.

6) At the end of the abstract the authors state: "Our theory offers a way to design an experimental geometry that is better suited for probing the phenomenon and to investigate the out-of-plane dynamics of ballistic quasitwo-dimensional materials." However, I do not see any discussion of this in the main text. Do the authors simply mean that one should change the aspect ratio L/W to see clearer oscillations, as in Fig. 3?

We indeed referred to the aspect ratio, however upon revisiting the manuscript we have identified this statement as hard to interpret, and therefore removed it.

- We clarified the relationship to the Kubo theory, in particular, explaining that Kubo theory is correct and that our approach extends that understanding by identifying the phenomenon as a consequence of the open trajectories and identifying the sample boundary scattering as the main mechanism limiting the intensity of the magnetoresistance oscillations.
- We have reorganized the derivation to make it more universal and straightforward to follow. In particular, we demonstrated that in a general setting the magnetoresistance response is described by the independent behavior of in-plane trajectories.
- We have restructured the discussion, and clearly state that it is the sample geometry that likely determines the magnetoresistance oscillations amplitude.
- We identified and included earlier works observing related phenomena in crystalline gallium in Ref 14 as well as a qualitative explanation in Ref 13.