Hidden Charge Orders in Low-Dimensional Mott Insulators

Round 1

Reviewer 1 Report

The first part of the manuscript presents an overview about hidden order and nonlocal order parameters to describe zero-temperature phases in low dimensional fermionic insulators, including the Extended Hubbard Model. The second part is devoted to the study of some more general models with long-range interactions, such as dipolar, screened and unscreened Coulomb ones.

 

The first part of the paper is a review of already known results, which are helpful for setting the background to understand the second and more original section. The latter presents some interesting and not obvious results about some models with long-range interactions, a topic which is of particular interest in recent times.

 

However, the paper has some weaknesses that I list below. For these reasons I believe the paper could be accepted for publication only after some revisions are made.

 

I have first some general comments to submit to the authors.

-      One of the most striking and characteristic properties of topological non-trivial phases is the appearance of zero-energy edge states when open boundary conditions are used. This issue is mentioned in the opening presentation part (sect. 1.4), but then it is no longer taken into consideration when discussing long-range models. A discussion of this case would be at order, also because they appear to have unusual properties in other long-range models (see next comment).

-      The effect of long-range interactions for one-dimensional chains of fermions and/or spin models has been considered in many papers such as:

P. Hauke, F. M. Cucchietti, M. Müller-Hermes, A. Bañuls, J. I. Cirac, M. Lewenstein, New J. Phys. 12, 113037 (2010) 


D. Peter, S. Müller, S. Wessel, and H. P. Büchler, Phys. Rev. Lett. 109, 025303 (2012) 


D. Vodola, L. Lepori, E. Ercolessi, G. Pupillo, New J. Phys. 18 (2016) 0150001 


I. Frérot, P. Naldesi, T.  Roscilde,  Phys. Rev. B 95, 245111 (2017)

In all these works, it has been emphasised that truly long-range interactions have striking effects on the phase diagram of the model, deforming it in a non-trivial way with respect to the short-range case. The effects include the disappearance of some phase transitions, edge states that acquire a mass, unusual decaying of correlation functions in massive phases. 

It seems to me that the examples presented in the manuscript never reach such intrinsically long-range regime. It would be very interesting to understand whether: i) the results of the paper are robust when entering such a regime and ii) granted so, analogous behaviours can be found in the considered long-range Hubbard-like models.

 

Then, I would like to signal out some smaller points:

-      The title is very generic and does not convey the true content of the paper.

-      Sect. 1.1. is very concise: this is necessary since it contains a collection of well-known results. Thus, it would be better to introduce only the necessary notions and notation: I would avoid for example to introduce the dual field $\theta$ and write Hamiltonian (3) directly in terms of the momentum field $\Pi$.

-      One should define more explicitly the action of the P and T symmetries.

-      The last sentence of sect. 1 is about the onset of d-wave superconductivity. I think this comment deserves a better justification. 

-      The models considered in sect. 2 are at half fillings. Can the authors add some comments about different fillings?

 

Finally there are some typos, such as the question mark after ref. 13 in line 94; a space missing after “(SWD),” in line 113; a comma instead of a full stop at the end of the sentence in line 183. 

 

Author Response

We thank the referee for the careful reading of the manuscript.

First we apologize for a possible misunderstanding along the submission process: the article is an invited review article for the special issue, and not an article reporting substantial new research. As such, the results presented in the last section are preliminary, representing a possible application of the method discussed in the review which needs to be further addressed by future work. 

We have added some words along the text to render more clear this aspect.

Coming to the specific comments of the referee:

1) The example discussed in the final section does not exhibit an insulating phase with non-trivial topological properties. The only phase with non-local nature is the Mott insulating phase with non vanishing charge parity, while the other two insulating phases (BOW and CDW) are characterized by local order.

2) We have added the suggested references about dipolar and long ranged interactions. We have specified in more details what we expect adding further sites and reaching the long range limit based on bosonization predictions. 

3) We resolved for the present title since the review aims at describing in terms of hidden orders the different types of Mott insulating phases induced by interaction in strongly interacting systems.

4) We have defined more explicitly the action of T and P

5) We have completed the sentence about d-wave superconductivity which was missing a line in the previous version.

6) Since we were interested in describing the possible different insulating phases, we did not discuss the non-insulating regimes away from half-filling

Reviewer 2 Report

The paper of Fazzini et al. deals with the characterization of the insulating phases of fermionic 1D models via non-local operators,
parity and/or string charge operators. In the case of a finite fractional non local parity charge order is capable of characterizing
also some two dimensional Mott insulators, both in the fermionic and the bosonic cases.
Through DMRG numerical analysis the robustness of both hidden orders at half-filling in the 1D fermionic
Hubbard model extended with long range density-density interaction is verified. The results obtained including
several neighbors in case of dipolar, screened and unscreened repulsive Coulomb interactions, confirm
the phase diagram of the standard extended Hubbard model. Besides the trivial Mott phase, also the
bond ordered and charge density wave insulating phases appear to resist to longer ranged interaction.
The paper is well written and in the first part offers a nice review of the previous work by the authors.
1. To render the paper more accessible to readers it would be worth to mention physically these operators how can be measured in experiments and which type of systems are more reliable to do this.
2. As for the long range interaction, have the authors checked that the results do not sensibly depend on the choice of the screening coefficient? Have they checked what happens using a different criterion?

3. For readers that are non familiar with bosonization it would be good to quote some review.
The paper can be accepted once the above comments are taken into account.

Author Response

We thank the referee for the careful reading of the manuscript.

First we apologize for a possible misunderstanding along the submission process: the article is an invited review article for the special issue, and not an article reporting substantial new research. As such, the results presented in the last section are preliminary, representing a possible application of the method discussed in the review which needs to be further addressed by future work. 

We have added some words along the text to render more clear this aspect.

Coming to the specific comments:

1) There are already experiments capable of measuring both spin and charge non-local operators in cold atoms systems by means of in-situ imaging techniques (see for instance our references [7], [8]). We have now mentioned this explicitly.

2) In case of long range interaction with a power law decay with exponents alpha, it is expected that adding further sites could not qualitatively change the phase diagram in case alpha is greater than one (see for instance ref. [9]).

In case of the screened Coulomb potential, we compared it to the dipolar interaction case (which has alpha=3) by ensuring that the first neglected terms were of the same order of magnitude in both cases. We expect that --for those lambda values for which the results of dipolar and screened interaction are comparable in the above sense-- the results still hold when the number of included neighbors tends to infinity. For instance, this should be the case both for $\lambda =0.6$ and $\lambda=0.7$ examined in the paper.

We added some comments on this point in the conclusions

3) a couple of review books on the bosonization method is already indicated in the references ([9,11]). We added a reference to them along the text when reviewing the bosonization results.

Reviewer 3 Report

This manuscript by Fazzini and Montorsi studies the characterization of phases
in a one-dimensional model of interacting fermions with long-range interactions,
in extension of work for the extended 1d Hubbard model. A particular focus is given
to the various types of nonlocal order parameters.

After a good introduction into the possible phases of 1d models with a charge
and a spin channel based on field theory and of the behavior of various nonlocal
order parameters, the authors focus on a set of specific models with longer-ranged
interactions. They used density-matrix renormalization group simulations to
demonstrate that in all cases a sequences of a Mott-insulating phase, a bond-order
wave phase and a charge density wave phase exists as a function of the strength
of the longer-ranged interactions. In praxis, the authors cutoff the range of the interaction
at a finite value.
 
The physics addressed here is timely (see, e.g., the experiment Ref. 8) and the general
interest in nonlocal order and symmetry-protected phases. In its present form, I
can, however, not recommend this paper for publication.

My main worry concerns the quality of the numerical results. The authors choose
to use periodic boundary conditions (PBCs) for which DMRG is much harder to converge.
Moreover, only results for a single system size are shown, leaving doubts on the
robustness as L increases.

The authors should clarify why PBCs are used. At least for one model and quantity,
an analysis of the numerical data quality should be presented. Generally, one
needs to use as many sweeps as necessary and 6 or 8 sweeps may not suffice.
Moreover, and more importantly, the dependence of the numerical results on
the number of DMRG states needs to be illustrated to provide an estimate of
numerical errors.

Then, the L-dependence needs to be addressed by at least plotting results for
L=16,24,32 on top of each other, but preferably also for larger L. Otherwise
one cannot trust the significance of the results.

Further comments:

- Throughout the text and in the figures and captions, the authors should be
consistent in their uses of units (i.e., U=4t instead of U=4).

- References to the DMRG literature are missing. Which specific algorithm
for PBCs was implemented?

- References on symmetry-protected phases in 1d would seem appropriate here
  (see, e.g. work by Pollmann, Turner, Wen and orthers).

- Line 55: I suggest to replace "Whereas"by "However"

- Line 94: something is wrong in the references: "[13?--15]"

- Caption of Tab 1: perhaps "staying "should be replaced by "standing"?

- Line 240, 241: One should not show data that are possibly not converged.
The severeness of finite-size effects should be illustrated by comparing
results for different system sizes.

Author Response

We thank the referee for the careful reading of the manuscript.

First we apologize for a possible misunderstanding along the submission process: the article is an invited review article for the special issue, and not an article reporting substantial new research. As such, the results presented in the last section are preliminary, representing a possible application of the method discussed in the review which needs to be further addressed by future work. 

We have added some words along the text to render more clear this aspect.

Coming to the specific points:

1) We prefer to use PBC in order to avoid boundary effects, that are quite relevant in the computation of non-local order parameters (see for instance our refs. [3],[14] and references therein). We have chosen the number of DMRG states in such a way that the truncation error is of the order 5x10^-6. We also verified that 6 sweeps are sufficient to reach convergence in the case of interaction truncated to third nearest neighbors (R_T=3), while we had to use 8 sweeps for R_T=5. 

2) A finite size scaling analysis was beyond the aim of the present work, that should be regarded as a review. We have shown data at fixed and quite large length as preliminary results about a possible application of the tools described in the review.

3) We have made the notation consistent throughout the text.

4) Some references to the DMRG literature have been added.

5) We have expanded the paragraph on topological insulators in order to better address their connection with symmetry protected topological phases, and added the relevant references for the latter.

6) We have fixed some typos, among which those indicated by the referee.

7) As already mentioned in the manuscript, in the case of R_T=5 it is more difficult to obtain very accurate results. However, those reported in figure 7 are accurate. The cited possible non-convergence of DMRG algorithm did refer to points not reported in the figure: since they are missing, their actual value could possibly change the apparent shift observed in the transition line. 

Round 2

Reviewer 3 Report

The authors addressed the technical comments from my previous report, however,

I don't find their answer regarding the quality of the numerical data convincing (points

1 &2 from their reply).

Regardless of whether this manuscript is intended to be a review or an original

article, it should report on finalized research. I feel rather hesitant to recommend

a manuscript for publication that itself states that the results are "preliminary".

In particular, a review should not talk about work in progress.

I am therefore not in a position to recommend this work for publication.