Many-body localization of spinless fermions with attractive interactions in one dimension

1- the manuscript offers a a very objective analysis of the numerical results on state-of-the-art system sizes for exact diagonalization of disordered quantum many-body systems;
2- it convincingly shows that no inverted many-body mobility edges are to be found in the full spectrum of the s=1/2 XXZ model in a random longitudinal field.

1 - a fundamental symmetry of the model mapping the attractive onto the repulsive side of the finite-energy phase diagram is overlooked. Relatedly, in light of this symmetry the absence of an inverted mobility edge in the special case of the SU(2) invariant system (both attractive and repulsive) has already been reported previously in the literature (even by some of the authors of this manuscript);
2 - the section of the paper concerning the ground-state study is rather qualitative, focused on methods (DMRG study of the occupation-spectrum discontinuity, FRG study of the conductance), and it does not add much to the known physics of the model in question.

This manuscript reports on the study of the localization properties of the one-dimensional S=1/2 XXZ model in a random longitudinal field with a flat-box distribution -- or, equivalently, of 1d spinless fermions with nearest-neighbor interactions and a random on-site chemical potential. These models have been abundantly studied in the recent past as paradigmatic realizations of the many-body localization (MBL), with a special focus on the XXX (or the SU(2) symmetric) limit (colloquially known as the "standard model" of MBL). The main purpose of the present manuscript is to explore the full anisotropy axis Delta of the XXZ model, encompassing both the antiferromagnetic (repulsive, V>0) regime as well as the ferromagnetic (attractive, V<0) one - using the magnetic (fermionic) language, respectively. The authors claim that the literature has mostly focused on the antiferromagnetic/repulsive side, but this statement should be revised. As a matter of fact, as pointed out e.g. in P. Naldesi et al., SciPost Phys. 1, 010 (2016), on average over disorder the particle-hole symmetry at half filling maps the physics of the Hamiltonian H(V) onto that of -H(-V). Therefore the many studies which inspected the localization properties of the *entire* energy spectrum of the XXZ model - and many of them are cited by this manuscript - automatically ventured both the attractive and the repulsive case. In this respect, the special case V/t=-2 (corresponding to the XXX model) has already been abundantly studied -- and particularly so by some of the authors in Ref. 25, for what concerns the behavior of the "occupation-spectrum discontinuity". Hence one of the main results of the manuscript, namely the absence of an inverted many-body mobility edge on the attractive side (namely of the scenario in which, upon increasing the energy density from the ground state, the system goes from extended to localized), is not per se entirely new, because such an absence was already abundantly reported in the phase diagram of the XXX limit. Nonetheless the manuscript goes beyond this particular limit, and convincingly shows that inverted mobility edges are not to be found in the energy-disorder phase diagram of the random-field 1d XXZ model, not even in the regime -2 < V/t < -1 in which the ground state remains an extended Luttinger liquid even for a finite disorder strength.
The paper is well written, and it delivers a very objective analysis of the numerical results -- which, as is well known in the context of MBL, are fatally limited to small system sizes when looking at finite energy densities. Moreover the manuscript also offers further elements about the ground-state phase diagram in the anisotropy/disorder plane, which has already been the subject of several studies in the past -- in particular density-matrix-renormalization-group (DMRG) results show that the "occupation-spectrum discontinuity" exhibits a significant minimum in the Luttinger-liquid phase, although this feature is not really used by the authors to explicitly estimate the phase transition from the LL phase to the localized (Bose glass) phase, and not much is said about its supposed critical behavior; and the authors also add functional-renormalization-group (FRG) results about the ground-state conductance of the model, in semi-quantitative agreement with very recent estimates of the phase boundary (Ref. 64). It is fair to say that the ground-state segment of the manuscript has more of a methodological taste - the manuscript shows that the "occupation-spectrum discontinuity" may be helpful qualitatively in the discrimination of extended vs. localized phases in the ground state; moreover it probes the effectiveness of the FRG approach - but it is also fair to say that it does not add too many new ingredients about the physics of the model as such. In this respect I would also like to notice that the authors omit to cite/comment the first bosonization study (to my knowledge) of the ground-state phase diagram of the model in question, namely Doty & Fisher, Phys. Rev. B 45, 2167 (1992); and a further DMRG study, Urba & Rosengren, Phys. Rev. B 67, 104406 (2003), reporting a phase diagram which seems to be in substantial disagreement with the one reported by the studies (Refs. 28 and 64) cited by the authors.
The most interesting part of the manuscript is certainly the finite-energy study, albeit with the slight limitation in novelty already mentioned above. Given the elements at hand, it is difficult for me to formulate a firm recommendation in favor or against publication in SciPost Physics; indeed, this journal is very young, and its standards are still being set. Hence I leave to the Editors the decision about whether such standards are being met, knowing that the manuscript contains solid results, but may generally lack novelty or have locally an incremental character.

Beyond the general remarks and the explicit requests already formulated in the above report, I have a series of more localized remarks/requests that the authors may want to consider:
1) I personally find Fig. 1 uselessly confusing: it overlays two scenarios on the same figure - and it takes a while (at least to me) to sort them out correctly - but then the rest of the manuscript completely rules out the scenario no. 1 beyond any reasonable doubt, nor has there been any microscopic model (to my knowledge) which is theoretically proven to exhibit such a scenario so far. Hence why should the authors create so much tension around a physical scenario - that of an inverted mobility edge - which is a priori not so particularly likely or justified?
2) I find the discussion of the "occupation-spectrum discontinuity" in Sec. II slightly confusing. The authors qualify the presence of such a discontinuity as evidence of "Fock-space localization" - meaning that, if the ground state is a unique Fock state in the single particle basis offered by the natural orbitals, as is the case in the non-interacting limit, then it is perfectly Fock-space localized. This means that, as the authors state, a Fermi liquid in d>1 would be localized, while, in terms of physical properties, such a regime would be rather qualified as extended and conducting. The fact that extended phases are characterized by a large delocalization in the Fock space built around the natural orbitals seems to be a special property of one spatial dimension, in which interacting Fermi liquids with a well defined Fermi surface do not exist. If one were to investigate localization phenomena for interacting fermions in higher dimensions using the same concept of localization in Fock space built upon the natural orbitals, what may one expect? A finite "occupation-spectrum discontinuity" in both the extended (metallic/Fermi liquid) and the localized phases, and maybe a vanishing discontinuity only at the transition?
3) The same Sec. II contains incorrect statements concerning the distribution of the adjacent gap ratio r: it states that r follows a Poisson distribution in the MBL phase, which is incorrect because the Poisson distribution pertains to the gap itself, while r follows the distribution 2/(1+r)^2, with an average of 0.386... (see Oganesyan & Huse, Phys. Rev. B 75, 155111 (2007)). To state that r has "a Gaussian orthogonal ensemble" does not mean anything to me: one should say that in the Gaussian orthogonal ensemble the average value of r is found to be 0.5295... (see again Oganesyan & Huse). The statements should be amended and the correct literature should be referenced.
4) Why do the authors use the arithmetic mean for the disorder average of the occupation-spectrum discontinuity in the ground-state study, and the geometric mean in the finite-energy study? The rationale of this choice should be clearly explained.
5) In Fig. 5, I would have expected the occupation spectrum discontinuity to be nearly zero everywhere in the range [-2,2] at zero disorder (namely the Luttinger-liquid regime). Why is it not so?
6) A couple of sentences may need some editing (see "an, in principle, exact solution" and "the question what the universality class of the delocalization-localization transition is models such as ours is still a topic of ongoing research").