Fermi-liquid corrections to the intrinsic anomalous Hall conductivity of topological metals

Dear Editor,
Thanks for sending us the two referee reports, which we carefully read and whose valuable comments and suggestions are incorporated into the revised version we are submitting here. It took us some time to extend the appendices to meet the Referees’ requests, particularly Appendix C.2, which is a crucial result in the multi-band generalisation of Landau’s Fermi liquid theory in the presence of a non-trivial topology. While it may appear lengthy and cumbersome, it is actually a straightforward application of perturbation theory for non-interacting electrons. We hope that the revised version will meet the expectations of both referees.

Yours sincerely,
Ivan Pasqua and Michele Fabrizio

Reply to the first Referee

First, we would like to express our sincere gratitude to the Referee for taking the time to thoroughly review our manuscript and providing us with such precise suggestions that have significantly enhanced the clarity of our work.
In the following, we will address each point raised in the report in detail.

I.1) Following Referee’s suggestion, we replace “already” with “naturally” and erase “of topological metals” from the abstract. Specifically, we now write “Furthermore, it demonstrates that such corrections are naturally accounted for by Landau's Fermi liquid theory, here extended to the case in which coherence effects between bands crossing the chemical potential and those that are instead away from it may play a crucial role, as in the anomalous Hall conductivity, …”, since, following a request by both Referees, we have substantially extended the Appendix, in particular adding Appendix C where we present a detailed microscopic derivation of Fermi Liquid theory in the realistic case of metals with many bands, only some of which cross the chemical potential, which is necessary to discuss topology. This is also the reason why it took us some time to resubmit the manuscript.

I.2) In the revised manuscript, we have delved deeper into the crucial work by Chen and Son, both in the Introduction and at the conclusion of Section 3. In their research, Chen and Son derive a linearised kinetic equation for a Fermi liquid with a Berry curvature. They identify corrections to the anomalous Hall conductance that they attribute to an electric dipole moment carried by the quasiparticles. On the contrary, the Fermi liquid corrections we derive within our formalism, as highlighted by the Referee, merely reflect the distinction between the static and dynamic limits of the linear response functions. While we could not find a straightforward physical argument to support this, we think it is possible that our findings align with the interpretation proposed by Chen and Son.

II.1) We agree with the Referee that the idea of discussing Fermi liquid theory from an ab-initio Hamiltonian that includes an infinite number of bands initially appears somewhat peculiar, even though the primary focus is on bands crossing the chemical potential and those responsible for the non-trivial topology. Nevertheless, the formalism presented in Appendix B is highly general and serves as the foundation for deriving Landau’s Fermi Liquid theory in Appendix C. Surprisingly, we found that this derivation appears to be valid regardless of the number of bands, particularly in the context of the anomalous Hall conductivity discussed in Appendix C.2. In fact, we thoroughly checked the calculations in that Appendix, which are lengthy and tedious, but ultimately consist solely of manipulating perturbation theory for non-interacting electrons described by a Hamiltonian where, see Appendix B, the Matsubara frequency just plays the role of an additional Hamiltonian parameter. Such calculations are commonly performed in all works that discuss the topological properties of solids, assuming the absence of electron-electron interaction, in which case, too, the formalism remains general and independent of the number of bands.

II.2) We revised the statement at the end of Sec. B, as well as the beginning of the Conclusions section, to clarify the point raised by the Referee. We believe, in fact, that equations (8), (9) and (10) have a more general validity, consistently with what we write above in point II.1). Indeed, it might well happen that an occupied band has still a topological character and thus contributes to Eq. (8) but with a non-quantised value because of (10), i.e., because of the interaction with the quasiparticles at the Fermi surface. This result is not wrong, as one might believe, since the contribution of the occupied band becomes quantised again if no band crosses anymore the chemical potential. Put it differently, we believe that our results, equations (8) to (10), constitute a generalisation of Haldane’s results, which, e.g., would imply that if the band crossing the chemical potential is not topological while an occupied band is, yet the anomalous Hall conductivity is not necessarily quantised.

II.3) As discussed earlier, in the revised manuscript, we have included Appendix C, which presents a formal and detailed microscopic derivation of Landau’s Fermi liquid theory for multiple bands. This derivation is a novel result that we believe is significant and surpasses the standard single-band case. We want to emphasise that, apart from the standard Fermi liquid assumptions, we make only one additional assumption: that a scattering process between distinct bands involving the dynamic correction to the current does not preserve phase coherence and averages to zero when summed over frequency, momentum, and band indices. This point is extensively discussed in Appendix C.2. Briefly, an interband transition induced by the dynamic correction to the current is a scattering process due to interaction between a quasiparticle-quasihole pair on two distinct bands at finite , and an intraband pair at on the Fermi surface. The phase coherence between the two pairs at different frequencies is not guaranteed, rather the contrary. Therefore, we argue that, whenever one has to sum over frequency, momentum and bands, this interaction-induced interband transition yields full decoherence and thus averages at zero. On the contrary, in all quantities that catch just the singularities at that arise from the discontinuity of the phase of the Green’s function, see Appendix C.2 for the case of the anomalous Hall conductivity, the above pairs are both at zero frequency and phase coherence is maintained.



We once again express our deepest gratitude to the Referee for the valuable feedback. We sincerely hope that the revisions address all concerns satisfactorily.



Reply to the second Referee

First, we would like to express our sincere gratitude to the Referee for taking the time to carefully review our manuscript and providing us with such precise suggestions that have greatly aided us in enhancing the presentation of our results.
In the following, we will address each point raised in the report in detail.

1.) In the revised manuscript, we failed to properly cite the work by Chen and Son. We have rectified this error and now give full credit to their work, as evident in the revised Introduction and the conclusion of Section 3. Nevertheless, despite the possibility that the corrections Chen and Son identify, which they attribute to a quasiparticle electric dipole, and our own, which stem from the distinction between static and dynamic limits in a metal, are related, we were unable to find a physical argument to support this claim.


Concerning the requested changes:

1.) We agree with the Referee, and apologise for not having emphasised the natural connection with gapless quantum spin liquids. We now mention the relationship in the Conclusions section and cite the relevant review pointed out by the Referee.

2.) We perfectly agree with the Referee and modified accordingly that sentence.

2.a) Since the same question was raised by the first Referee, the revised version includes a new Appendix C. In this appendix, we present a detailed microscopic derivation of Landau’s Fermi liquid theory. This theory generalises the one by Nozières and Luttinger when the interaction between bands that cross the chemical potential and those that are either fully occupied or empty plays a crucial role. This is precisely the case of the anomalous Hall conductivity. This is also the reason why it took us time to resubmit the manuscript. Since this derivation is noteworthy on its own, we refer to it more extensively in the text. In fact, we already mention it in the abstract, where we borrow an enlightening sentence from the Referee, with their permission. We emphasise that the only additional assumption that we make besides the standard Fermi liquid ones is that a scattering process between distinct bands that involves the dynamic correction to the current does not maintain phase coherence, and thus averages at zero upon summing over frequency, momentum and band indices. This point is discussed at length in Appendix C.2. Briefly, an interband transition induced by the dynamic correction to the current is a scattering process due to interaction between a quasiparticle-quasihole pair on two distinct bands at finite , and an intraband pair at on the Fermi surface. The phase coherence between the two pairs at different frequencies is not guaranteed, rather the contrary. Therefore, we argue that, whenever one has to sum over frequency, momentum and bands, this interaction-induced interband transition yields full decoherence and thus averages at zero. On the contrary, in all quantities that catch just the singularities at that arise from the discontinuity of the phase of the Green’s function, see Appendix C.2 for the case of the anomalous Hall conductivity, the above pairs are both at zero frequency and phase coherence is maintained.

3.) We thank the Referee for pointing to our attention relevant works that we originally missed, and which we cite in the revised version.


We once again express our gratitude to the Referee for providing valuable feedback. We sincerely hope that the revisions address all concerns satisfactorily.