|As Contributors:||Christoph Karrasch|
|Submitted by:||Karrasch, Christoph|
|Submitted to:||SciPost Physics|
|Subject area:||Condensed Matter Physics - Theory|
We study the finite-energy 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 Luttinger-liquid phase persists at weak disorder in the ground state, which is a well-known result. We revisit the ground-state phase diagram and show that the recently introduced occupation-spectrum discontinuity computed from the eigenspectrum of one-particle density matrices is noticeably smaller in the Luttinger liquid compared to the localized regions. Moreover, we use the functional renormalization scheme to study the finite-size dependence of the conductance, which resolves the existence of the Luttinger liquid as well and is computationally cheap. Our main results concern the finite-energy density case. Using exact diagonalization and by computing various established measures of the many-body localization-delocalization transition, we argue that the zero-temperature Luttinger liquid smoothly evolves into a finite-energy density ergodic phase without any intermediate phase transition. As a consequence, the full energy-density versus interaction strength diagram has a pronounced asymmetry between the attractive and the repulsive side.
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
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.
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
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
In this paper, the authors explore the asymmetry between repulsive and attractive interactions in the finite-energy phase diagram of disordered interacting Fermions. In particular, they focus on the evolution of the mobility edge separating the ergodic and many-body-localized (MBL) phases with increasing energy in the case of attractive interactions (a less-studied regime of parameter space in the previous literature). To this end, they study a 1D model of spinless Fermions with nearest-neighbor attractive interactions (V) and on-site 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 low-energy (where a delocalized Luttinger liquid phase is stable at intermediate V scale) and high-energy (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 delocalization-MBL 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.
A slight extension of the physical aspects in the Conclusions section, as detailed in the report.