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

1. Non-perturbative results for Hall conductance in quantum many-body problem
2. Interesting new strategy to access ongoing questions in strongly interacting topological insulators (by considering them as un-doped Fermi liquids)
3. Paper well structured in firm results (Secs. 2&3) and interesting conjectures (conclusions)

1. Technicalities are a bit cumbersome (I would have found a case study for a continuum system with full rotational symmetry more intuitive than the lattice model)

The paper on Fermi liquid corrections by Pasqua and Fabrizio addresses a very topical and at the same time long-standing question of strongly interacting anomalous Hall metals by carefully taking into account corrections due to residual Fermi liquid interactions.

If correct, this work provides a substantial step forward beyond previous results and may allow to connect the Berry Fermi liquid theory with the insulating counterpart. As such, I would agree with the authors that the manuscript ticks the boxes of groundbreaking theoretical discovery/theoretical breakthrough.

However, I have a multitude of remarks/questions to be answered before I can take a final decision.

In the answer to the referees, could the authors please elaborate on the following question?

1. Can the authors comment on the insufficiency of previous works, in particular of the kinetic equation based approach by Chen and Son, Ref [9]? What is missing there?

(For requested changes, please see below)

1. Regarding the discussion part: It is well know that the discrepancy between Chern and winding number can occur in the presence of fractionalization, e.g. quantum spin liquids. These can also be be understood by sending certain Fermi liquid parameters to -2 while keeping other FL parameters finite, cf RMP 89, 025003 (2017) and references therein. I feel this analogy should be discussed more clearly in the paper.

2. I find the statement that the "intrinsic anomalous Hall conductivity ... is... genuinely a property of the quasiparticle Fermi surface" very dangerous. Already the non-interacting, clean Berry curvature contribution can be converted from Fermi surface to Fermi sea contribution by a single partial integral. In fact, the Berry curvature is really a manifestation of interband coherence (see Eq. (9)).

2a) Above all, I ask the authors to explain to the readers why Fermi liquid theory (which is geared to only make statements about the Fermi surface and its vicinity) can be trusted in this calculation of interband effects.

2b) Beyond that, I propose to adapt the wording in the introduction.

3. The referencing of works on topological Green's function zeros is rather incomplete. In particular, I find that the following articles from various different groups and using a variety of different techniques are important for this work but missing:
a. PRB 90, 060502(R) (2014),
b. PRL 133, 126504 (2024),
c. PRL 133, 136504 (2024),
d. arXiv:2405.08093.

Ask for minor revision

The authors pose and analyze a very interesting and relevant question about Fermi-liquid corrections to the intrinsic anomalous Hall effect in metals. About 20 years ago, Haldane has shown that in the noninterating Fermi gas, a fractional part of the Hall conductivity, given by the integral of the Berry curvature over the Fermi volume, is in fact a Fermi surface property as it reduces to a Fermi surface integral of the Berry connection. It is therefore tempting to assume that the same holds true in the interacting Fermi liquid because at the Fermi surface the quasiparticles and their wave functions, determining the Berry connection, remain well-defined. The authors challenge this seemingly natural assumption and argue that the residual interaction between quasiparticles generate nontrivial (vertex) corrections to the Haldane formula for the Hall conductivity. This result may have
serious implications not only for topological materials near the Mott transition, as the authors discuss, but also for the first principles calculations of the Hall conductivity in usual ferromagnets. As the Landau parameters in metals are typically not small, the standard Berry curvature based results for the intrinsic contribution to $\sigma_H$ become somewhat questionable and should be reconsidered. Therefore, the presented results, if correct and presented a bit more clearly (see
below), can justify publication in SciPost in accordance with the last two items in the list of expectations (items 3 and 4 as indicated by the authors).

I personally find the main statement of this work very plausible physically. I like very much a simple analogy to the Drude weight in metals. When computing linear transport coefficients, the external current vertices should be taken in the "nonequilibrium" $\omega$-limit, which makes the appearance of interaction (vertex) corrections unavoidable. Despite I tend to agree with the final result, I think there are several problematic points in the presentation and argumentation, which should be clarified before a possible acceptance.

I. General questions to the abstract and introduction

1. In the abstract, the authors state that the corrections they discuss are already accounted for by the "Landau's Fermi liquid theory of topological metals". However, it is not explained in the
text what this actually mean. Moreover, it seems this term does not even appear in the main text.

2. In the introduction, the authors quote, but do not discuss, the paper of Ref.[11] by Chen and Son. However, it looks very relevant to the present work. As far I can see, in Ref.[11] Fermi-liquid interaction corrections to the Berry phase expression for the Hall conductivity are discussed, calculated, and attributed physically to a "dipole moment of the quasiparticles". It is not clear to me whether it is exactly the same or not, but I think in such a situation a more detailed discussion of the relation to Ref.[11] is necessary.

II. Technical points which require clarification

1. I think the basic statement of the problem in Sec.2 should be explained better. What mean $\alpha, \beta$ indexes in eq.(1). I understand that the authors probably have in mind tight-binding or similar models, but what it could be in the real world. Should we understand $H_*(k)$ as a differential operator in real space, i.e. the Hamiltonian of a crystal in the k-representation defined on a unit cell with periodic boundary conditions. If so, it has an infinite number of discrete eigenvalues for each k in the BZ -- the crystal's band structure. It makes a sense to indicate this more clearly, as considering infinitely many discrete eigenvalues (the bands) at each k looks highly unusual in the context of the Fermi liquid theory.

2. If the above understanding is correct, then it is probably also correct that the physical significance have only the eigenstates of $H_*(k)$ which belong to the Fermi surface, while the
overwhelming majority of states/bands that appear in the present formalism, but have energies away from the Fermi energy, are strictly speaking unphysical. In my opinion, this also
requires a clarification, especially in connection to the final formulas (8)-(9) for the Hall conductivity, which explicitly involve integration over the unphysical states. What looks to me
quite disturbing is that these states appear as eigenstates of a fictitious eigenvalue problem with Hamiltonian $H_*(k)$. Possibly these fictitious states somehow disappear at the very
end, as it apparently happens in the specific toy example in Sec.3. I believe this issue is serious enough to be discussed in some length, especially in the situation when the general demonstration of this fact is absent.

3. It is in general a very good idea to present more technical details and rigorous arguments/proofs in appendices. In particular, Appendix B is aimed at justifying the approach used in Sec.2 and detailed in Appendix A. In my opinion, the most delicate point in the whole paper is the calculation of the linear response (current-current correlation) function using the "pseudo-HF" Green function (3) with a frequency independent self-energy. Unfortunately, precisely this point remains unexplained. The justification relies on eq.(14), which is indeed obvious for n=l at the Fermi surface, but does not look trivial if n is not equal to l. For the off-diagonal terms the electron-hole excitations are gapped, which makes the corresponding contribution non-singular and therefore the arguments typically used in similar derivations, for example in Ref.[14], do not apply, at least straightforwardly. If eq.(14) and a formal justification of the applied methodology can indeed be "readily found", it should be presented. I believe this will allay most of my concerns.

After the requested clarifications and explanations, I would probably recommend this work for the publication. In general, the manuscript is sufficiently well written, cites the relevant literature, and satisfies (up to the above listed points) the standard criteria required for a good scientific work.

Ask for major revision