# One-dimensional extended Hubbard model with soft-core potential

### Submission summary

 As Contributors: Thomas Botzung Arxiv Link: https://arxiv.org/abs/1909.12168v2 (pdf) Date submitted: 2020-07-10 02:00 Submitted by: Botzung, Thomas Submitted to: SciPost Physics Discipline: Physics Subject area: Quantum Physics Approach: Theoretical

### Abstract

We investigate the $T=0$ phase diagram of a variant of the one-dimensional extended Hubbard model where particles interact via a finite-range soft-shoulder potential. Using Density Matrix Renormalization Group (DMRG) simulations, we evidence the appearance of Cluster Luttinger Liquid (CLL) phases, similarly to what first predicted in a hard-core bosonic chain [M. Mattioli, M. Dalmonte, W. Lechner, and G. Pupillo, Phys. Rev. Lett. 111, 165302]. As the interaction strength parameters change, we find different types of clusters, that encode the order of the ground state in a semi-classical approximation and give rise to different types of CLLs. Interestingly, we find that the conventional Tomonaga Luttinger Liquid (TLL) is separated by a critical line with a central charge $c=5/2$, along which the two (spin and charge) bosonic degrees of freedom (corresponding to $c=1$ each) combine in a supersymmetric way with an emergent fermionic excitation ($c=1/2$). We also demonstrate that there are no significant spin correlations.

###### Current status:
Editor-in-charge assigned

Dear Editor,
We apologize for the delay in the resubmission, and thank you and the referees for your patience. We also thank the referees for taking the time to read the manuscript carefully and for providing constructive criticism to improve it.

We appreciate Referee B for her/his positive remark: “ ... the work contained in this manuscript deals with an interesting and timely topic. Thus, the author’s findings, if substantiated, would certainly be suitable for publication in SciPost and be of interest to the community.” Unfortunately, Referee B does not recommend publication asking for analytical improvements and a better presentation. Referee A does not recommend publication due to “ Analysis of the numerical data is not complete and is not reproducible, due to missing details...”. Referee C also points out the lack of structure and the data presentation of the paper.

We understand the points raised by the referees. Thus we have restructured our paper to improve clarity (see list of changes) and answer all raised questions, as detailed in the different detailed answers. The presentation of our results may have been unclear to some readers. Therefore, we adapted our manuscript in several places following recommendations of referees (as detailed below). We now emphasize the classical limits and extent the numerical aspect by notably adding a new appendix section. We are convinced that our modified manuscript is suitable for publication in SciPost journal.

Sincerely,
Thomas Botzung, Guido Pupillo, Pascal Simon, Roberta Citro and Elisa Ercolessi

### List of changes

-We have added a completely new section, “Classical analysis” that considers the exact solution of the ground state in the classical limit. In this section, we show the different possible phases present in our model and demonstrate the crossover between CLLnn and CLLd.
-We have extended the section “observables” by adding more information about the structure factor, entropy, and low-energy degrees of freedom. In particular, we explicitly derive results of low-energy degrees of freedom in the classical limit.
-We have added a new Appendix, where we explain in detail how we extract the central charge from the Cardy-Calabrese formula.
-We have included a new figure in the main text and paragraph to confirm the crossover between CLLnn and CLLd in Sec 3.3. (see Eq. 14 and Fig.7)
-We have modified the outline and the conclusion to include the new section.

### Submission & Refereeing History

Resubmission 1909.12168v2 on 10 July 2020
Submission 1909.12168v1 on 27 September 2019

## Reports on this Submission

### Report

The manuscript has been improved during the revision. In particular, it contains some examples of finite-size and finite-bond scaling that makes the results reproducible and that allows the reader to make own conclusion based on the raw data.

1. My main concern in this new revised version is possible formation of the floating phase – Luttinger liquid phase with non-frozen (varying) wave-vector q. As I already pointed in my first review, there are numerous indication for it: (i) the structure factor has a peak at the intermediate value of k in Fig.12(a) and (ii) with increasing L the peak goes away from its commensurate value 2\pi/3.
The distance between two commensurate values 2\pi/3 and 3\pi/5 is only 0.0667\pi is the same order of magnitude as an error in numerically extracted value of q: \delta q\approx 2\pi/L\approx 2\pi/60=0.033\pi. This definition of error corresponds to an assumption “that L always contains an integer
number of clusters” mentioned on p.11 and the fact that correlation length always diverges.
These simple calculations implies that in order to clearly distinguished between the two scenarios - either direct transition between CLL_nn and CLL_d considered in the paper or the floating phase that I would expect – one has to go to the system sizes an order of magnitude larger than those presented in the manuscript. Let me be more explicit, to be able to measure, say, 10 points between two commensurate values 2\pi/3 and 3\pi/5 one has to do simulations on L>600 sites. Before these simulations are performed I do not see how one can choose one option over the other. So either this puzzle has to be resolved (e.g. numerically) or all possibilities have to be listed.

2. Critical system with (anti-)periodic boundary conditions, L>600 and c>~2 is quite unrealistic task. By contrast, in open systems where the bond dimension D is roughly equal to a square-root of the corresponding bond dimension for periodic one, such system sizes often can be accessed. But here comes the second puzzle – the authors write:
“Indeed, we did try with open boundary conditions, finding that edge effects arising from finite lattices kill completely the cluster formation.”
First, I 100% agree with the first Referee who suggests that this statement should appear in the main text.
Second, if cluster formation is a bulk process, it should appear with open boundary conditions as well. Of course, much larger system sizes or fixed (conformally invariant) boundary conditions might be necessary to overcome the edge effects. But this issue of “complete killing of the cluster formation” definitely requires systematic investigation, to exclude the possibility that CLL_d phase is a finite-size effect that will be ruined in the thermodynamic limit.

p.13 “Finally at strong V , k c is equal to π/3, which corresponds to the CLL_d phase” should be 2π /3

p.12 “the discontinuity in the energy between the CLL_nn”. The energy has to be continuous, do the authors mean discontinuity in its derivative?

Fig. 8. What is the meaning of “a possible extended critical region up to V ∼ 7”. The entire parameter space is critical isn’t it?

p.23 “Thus, for sure finite-size effects are strong resulting -as we will see- in an underestimation of the central charge.” Finite-size effects usually result in over-estimation of the central charge, this can be see n in panel 16(b), c decreases with 1/L. By contrast finite DMRG bond dimension indeed results in underestimated central charge.

In Fig.3 it is not clear how and where the TLL phase is separated from the TLLd phase.

Let me repeat what I already wrote in my first review. I think the problem itself is interesting and relevant. However, there are still important questions that remains open or have been overlooked in the current manuscript. Therefore I cannot recommend the publication of the manuscript before points 1 and 2 will be addressed.

• validity: high
• significance: good
• originality: good
• clarity: good
• formatting: perfect
• grammar: excellent

### Report

With their revised version, the authors have improved their manuscript and addressed some of the previous concerns. However, some issues remain open and new ones have been introduced by the revision.

The previous Report 1 pointed out some older references for Cluster Luttinger Liquids (CLLs, see Lecheminant et al., PRL 95, 240402 (2005); Roux et al., EPJB 68, 293 (2009); etc.). One might argue that this is not exactly the same model and thus not of direct relevance. However, the authors write in their reply:

| We have changed the reference in the abstract and added the suggested references in the introduction.

Neither of the two is actually true. If this is just an oversight, this should be fixed.

The second point that was raised previously concerns the issue of boundary conditions. The authors write at the beginning of section 2 and in Appendix A that they have used "antiperiodic" boundary conditions. In their reply, they write

| Open boundary conditions introduce boundary effects that can mask the cluster structure of the ground state in some regions of the phase diagram.

While it is true that open boundaries give rise to additional finite-size effects, they usually require much smaller numerical effort in DMRG, i.e., a much smaller value of $D$ is able to provide quantitatively accurate results than needed for (anti-)periodic ones. So, there is a balance to strike between numerical accuracy and additional finite-size effects. The authors write at the end of their reply to Report 3:

| Indeed, we did try with open boundary conditions, finding that edge effects arising from finite lattices kill completely the cluster formation.

This is a statement that in my opinion would merit integration into the manuscript. One might also wonder why antiperiodic boundary conditions are better than periodic ones. Is this related to the specific choice of filling $\rho=2/5$?

I have some further minor comments:
1- The first paragraph of the Introduction talks about field theories, but then the authors study a lattice model. While field theories do emerge as low-energy long-distance descriptions of lattice models, I think that an explicit statement on the relation would be useful for the reader.
2- The order of the references [52-55] in the Introduction and at the beginning of chapter 3 is strange: they start with a recent library [52] and finish with the original paper [55]. I recommend to restore chronological order (or separate method and implementation).
3- Line 3 of page 4 and line below Eq. (17): "sound speed" -> "speed of sound" (or "sound velocity").
4- What is "soft" about the hard cutoff of the repulsion at $r_c$? I think that the authors should explain the terminology "soft-shoulder" (or change it).
5- Typesetting "$r_c$" versus "r$_c$" should be unified.
6- Typesetting "CLL$_{\rm nn}$" and "CLL$_{\rm d}$" versus "CLL$_{nn}$" and "CLL$_{d}$" should be unified.
7- Page 9, section "Von Neumann entropy": it is strange to have no reference about these concepts until the line before Eq. (12).
8- On page 15, the authors talk about "a factor of 1/2", but I think that they mean an additive rather than a multiplicative constant.
9- In Fig. 11, the blue lines are difficult to distinguish from the black lines separating the different panels. In addition, the statement "our numerical data confirm that ALL sound speeds become the same" on the line below Eq. (17) is obviously not true for $v_s$.
10- The relation of the new Appendix B to the old Appendix A is not clear since they discuss very similar issues. I suspect that the Appendix A concerns the CLL$_{nn}$ phase while the Appendix B concerns the transition from the CLL$_{nn}$ to the CLL$_d$ phase. If this is correct, explicit statements to this effect would be helpful.
11- Even if this can probably be fixed during the production stage, I nevertheless suggest to fix the spelling of the chemical formula "SrCuO$_2$" in Refs. [11] and [14].
12- Ref. [17] duplicates Ref. [1].
13- Even if this can probably be fixed during the production stage, I nevertheless suggest to fix the name "Hall" in Refs. [21] and [22].
14- There are some further minor typographic errors (such as "L" versus "$L$" on line 6 of of the section "Von Neumann entropy", "CLuster" on line 6 of the caption of Fig. 3, "double occupied" versus "doubly occupied" [several instances], "U" versus "$U$" on the first line of section 3.4, etc.) that I hope can be fixed during the production process.

I recommend publication of the manuscript once the above issues have been addressed.

• validity: good
• significance: good
• originality: good
• clarity: high
• formatting: excellent
• grammar: good