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Manybody localization of spinless fermions with attractive interactions in one dimension
by ShengHsuan Lin, B. Sbierski, F. Dorfner, C. Karrasch, F. HeidrichMeisner
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Submission summary
Authors (as registered SciPost users):  Christoph Karrasch · Björn Sbierski 
Submission information  

Preprint Link:  http://arxiv.org/abs/1707.06759v2 (pdf) 
Date submitted:  20171010 02:00 
Submitted by:  Karrasch, Christoph 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
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Approach:  Theoretical 
Abstract
We study the finiteenergy density phase diagram of spinless fermions with attractive interactions in one dimension in the presence of uncorrelated diagonal disorder. Unlike the case of repulsive interactions, a delocalized Luttingerliquid phase persists at weak disorder in the ground state, which is a wellknown result. We revisit the groundstate phase diagram and show that the recently introduced occupationspectrum discontinuity computed from the eigenspectrum of oneparticle density matrices is noticeably smaller in the Luttinger liquid compared to the localized regions. Moreover, we use the functional renormalization scheme to study the finitesize dependence of the conductance, which resolves the existence of the Luttinger liquid as well and is computationally cheap. Our main results concern the finiteenergy density case. Using exact diagonalization and by computing various established measures of the manybody localizationdelocalization transition, we argue that the zerotemperature Luttinger liquid smoothly evolves into a finiteenergy density ergodic phase without any intermediate phase transition. As a consequence, the full energydensity versus interaction strength diagram has a pronounced asymmetry between the attractive and the repulsive side.
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Author comments upon resubmission
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 particlehole 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 groundstate 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 wellknown Luttingerliquid phase at T=0) and the analysis of a set of different measures of the delocalizationlocalization transition.
We agree that the portion of the manuscript that discusses the groundstate properties does not advance much the understanding of the zerotemperature 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 energydensity 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 manybody 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 BoseHubbard model as discussed in the introduction. Moreover, there are two theoretical studies of the 1D BoseHubbard 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 Fockspace localization. The beauty of studying the oneparticle 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 Fockspace localization is not exclusively a consequence of localization. Noninteracting systems are always Fock space localized and Fermiliquids are, at zero temperature, very weakly Fock space delocalized. Thus, observing Fockspace 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 Fockspace 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 Fermiliquid 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 occupationspectrum discontinuity.
5) On a finite system, even a clean LL will still have a finite OPDM occupationspectrum 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/highenergy density phase diagram for repulsive interactions.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 1 on 20171210 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1707.06759v2, delivered 20171210, doi: 10.21468/SciPost.Report.292
Strengths
1) Technically solid
2) Addresses a timely problem
3) Assists in the construction of a complete phase diagram in a model previously explored only in (relatively easy) extreme limits
Weaknesses
1) The presentation is elaborate on the technical side, and the discussion of physical intuition is somewhat lacking
2) The results are not particularly surprising
Report
In this paper, the authors explore the asymmetry between repulsive and attractive interactions in the finiteenergy phase diagram of disordered interacting Fermions. In particular, they focus on the evolution of the mobility edge separating the ergodic and manybodylocalized (MBL) phases with increasing energy in the case of attractive interactions (a lessstudied regime of parameter space in the previous literature). To this end, they study a 1D model of spinless Fermions with nearestneighbor attractive interactions (V) and onsite disorder at finite energy density. Using various numerical measures to distinguish the ergodic (delocalized) phase from the MBL phase, they demonstrate quite conclusively that the known limits of lowenergy (where a delocalized Luttinger liquid phase is stable at intermediate V scale) and highenergy (where the phase diagram is effectively symmetric under change of sign of V) are smoothly connected. The topology of the phase diagram is hence consistent with the scenario depicted in Fig. 1a, and the possibility of an inverted mobility edge (as in Fig. 1b) is excluded.
In my view this is a quite useful contribution to the hot topic of MBL physics, which bridges the gap between extensively studied limits and provides a more complete picture of the phase diagram. As far as I can judge, the numerical calculations implemented in this study (which incorporate several different methods) are solid. Besides the main physics result, they provide additional information of technical nature and potentially useful guidelines for future studies: for example, the authors discuss the limitations of the functional RG method, and verify that the adjacent gap ratio and the occupation discontinuity are better numerical probes of the delocalizationMBL transition than the variance of entanglement entropy.
In view of the above assessment, I'm in favor of publication in SciPost. However, there are a couple of aspects of the physics that I believe deserve a slightly more extended discussion:
1) The authors point out that the lack of inverted mobility edge is in contrast with studies of other models for dirty superfluids, such as Bosonic models with repulsive interactions. Can they elaborate on the fundamental distinction between the two types of system? Is there a regime where attractively interacting Fermions, in the presence of disorder, are expected to be effectively equivalent to such Bosonic systems?
2) What is the nature of the localized (MBL) phase at large negative V? is it qualitatively the same as the weakly interacting phase?
I believe that a discussion of these points in the "conclusions" section (where presently the discussion of physical interpretation is somewhat too brief compared to the technical aspects) will improve the presentation of the paper.
Requested changes
A slight extension of the physical aspects in the Conclusions section, as detailed in the report.