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

We thank the Referee for their comments on our work.

We are pleased that the Referee appreciates two of the main strengths of our work, i.e., its comprehensiveness and the presentation of conclusive evidence for the absence of an inverted mobility edge in the model of spinless fermions with attractive interactions in one dimension.

As for the two "weaknesses" identified by the Referee, we agree that the restoration of particle-hole symmetry after disorder averaging and the implications for the phase diagram should be discussed in our paper. However, different from what is alleged in the report, there is no final evidence in the literature for the complete absence of an inverted mobility edge even if previous work on the high energy portion of the XXZ chain with repulsive interactions is taken into account. The Referee makes frequent references to the XXX chain, for which indeed some quantities have been computed at all energy densities. However, the corresponding value at negative intertaction strength, V=-2t, sits at the edge of the delocalized phase in the ground state. The available results for the ground-state phase diagram (see, e.g., our Figs. 3 and 4a) suggest that any finite disorder leads to localization for V=-2t. Hence one wouldn't expect an inverted mobility edge there anyways and we therefore don't learn much from the results for XXX chains for this particular question. We therefore do not see what justifies the Referee's claim that "... (this result)... has already been reported previously in the literature...". A conclusive answer to the question of an inverted mobility edge requires a study if the full relevant parameter range (scanning the region on top of the well-known Luttinger-liquid phase at T=0) and the analysis of a set of different measures of the delocalization-localization transition.

We agree that the portion of the manuscript that discusses the ground-state properties does not advance much the understanding of the zero-temperature physics as such but it is nevertheless important to give a comprehensive picture and this section serves both as an introduction and motivation for the analysis of the finite energy-density properties. Moreover, we are quite excited about the results from functional RG for the conductance at zero temperature, which in our view invite future method improvements of FRG and applications to other disordered many-body systems.

The Referee further states, in various points in his/her report, that the absence of an inverted mobility edge is not surprising or expected. We reiterate that previous results for XXX chains make no implications for the presence of an inverted mobility edge on top of the Luttinger liquid (LL) phase in the -2t<V <-t regime.

Apart from the existence of the LL phase in the XXZ model with attractive interactions (which in the existing literature on MBL has not been taken into account), there is experimental evidence for the existence of an inverted mobility edge in the 2D Bose-Hubbard model as discussed in the introduction. Moreover, there are two theoretical studies of the 1D Bose-Hubbard model with disorder (new references 41 and 42) that also indicate the presence of an inverted mobility edge, which is thus not a question with a forgone answer and which likely exists in relevant models. Therefore, we disagree with the Referee's final assessment that our manuscript "...may generally lack novelty...".

Response to the specific comments and suggestions from the Referee's report:

1) We split Fig. 1 into two panels, separately showing the two scenarios, as suggested by the Referee.

2) MBL is characterized by localization in real space and by Fock-space localization. The beauty of studying the one-particle density matrix (OPDM) is that it gives access to both aspects of MBL, via the structure of natural orbitals and the occupation spectrum.

It is important to stress that Fock-space localization is not exclusively a consequence of localization. Non-interacting systems are always Fock space localized and Fermi-liquids are, at zero temperature, very weakly Fock space delocalized. Thus, observing Fock-space localization via the presence of an OPDM occupation spectrum discontinuity does not imply localization per se.

In physical systems in dimensions higher than one, the same phenomenology is expected in an MBL phase: localized natural orbitals and Fock-space localization (fully consistent with the results by Basko, Aleiner and Altshuler). See also a study of MBL in 2D that makes the same observations (Ref. 44).

A 3D interacting Fermi gas would have a transition from an ergodic phase (with a smooth OPDM occupation spectrum) to an MBL phase with a discontinuous occupation spectrum.

Note that a Fermi-liquid has a discontinuous occupation spectrum only at zero temperature, while any finite temperature leads to a smooth spectrum. Thus the scenario proposed by the Referee applies to T=0 only.

3) We thank the Referee for pointing out these inconsistencies regarding the discussion of the properties of the gap ratios vs the gaps themselves. We corrected the respective text and added the references suggested by the Referee.

4) We changed all figures to showing the arithmetic mean of the occupation-spectrum discontinuity.

5) On a finite system, even a clean LL will still have a finite OPDM occupation-spectrum discontinuity. Thus, in the clean case, it requires very large systems to see the correct LL behavior at k_F, where the slope diverges logarithmically. Hence it would be unrealistic to expect a vanishing discontinuity on system sizes with L=32 sites.

6) We thank the Referee for pointing out these issues. We improved the respective sentences.

(now referring to specific comments from the main report)

7) We added the references mentioned by the Referee on the ground state phase diagram ( Doty & Fisher, Phys. Rev. B 45, 2167 (1992); Urba & Rosengren, Phys. Rev. B 67, 104406 (2003), as well as Apel, Rice Phys. Rev. B 26, 7063 (1982) and slightly modified the discussion.

[We do not understand why the Referee states that the results of the DMRG study Urba & Rosengren, Phys. Rev. B 67, 104406 (2003) are in "substantial disagreement" with (old) Refs. 28 and 64. The LL phase is observed for the same range of interaction strength and differences in the critical field result from the different units and are within the error bars of these (early) DMRG simulations].

8) We added a discussion of the symmetries of the model that can be restored by disorder averages, also referring to P. Naldesi et al., SciPost Phys. 1, 010 (2016), and discussed the implications for the phase diagram at attractive interactions based on the knowledge of the finite/high-energy density phase diagram for repulsive interactions.