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

### Submission summary

 As Contributors: Thomas Botzung Arxiv Link: https://arxiv.org/abs/1909.12168v1 Date submitted: 2019-09-27 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. Dalmonte, W. Lechner, Z. Cai, M. Mattioli, A. M. L\"auchli and G. Pupillo, Phys. Rev. B 92, 045106 (2015)]. 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

### Submission & Refereeing History

Submission 1909.12168v1 on 27 September 2019

## Reports on this Submission

### Report

In the present manuscript the authors study an extended Hubbard model with nearest- and next-nearest-neighbour repulsion with equal couplings. The authors reports the appearance of the so-called cluster Luttinger liquid phases and discuss the transition between them.
The problem itself sounds very interesting and relevant, and after substantial revision the manuscript can be suitable for publication in SciPost. However at this stage the presentation of the numerical results is very poor and sometimes misleading. In particular, the very existence of the critical line between two types of CLL has not been justified. Can it be a crossover with varying size of the supercell (equivalently with varying k) as a function of coupling U and V? The possibility of the first-order transition suggested by the semi-classics has not been excluded as well. Analysis of the numerical data is not complete and in not reproducible, due to missing details (see comments below). Therefore I do not recommend the manuscript for publication in its current form.

My detailed comments and questions are listed below:

C0: I would be curious to see the plot of the wave-vector k as a function of U and V coupling constants. I am asking for something similar to Fig.4(a) but, perhaps with better resolution around V=1.5 and V>7; Since both CLL phases are critical one can easily imagine that at an intermediate value of the coupling constants, a competition between the two leads to a new state with an intermediate values of the wave-vector k. Of course, accurate analysis would require larger system sizes. But there is already a very good indication in favour of incommensurate values of q - in Fig.5(a) peaks for V=7.4 are off their commensurate values. Also, incommensurate peaks have been reported in Fig.10(a). However, finite-size scaling in panel (b) has been performed only for V<=4.5 and only with three data-points, which might not be sufficient in the presence of incommensurability. In this respect it might be instructive to do the calculation of the chain with L which is not multiple of 10.
C1. Fig.1: V=2U/3 does not go to zero; red line corresponds to semi-classical results, and it is not actual phase boundary, perhaps it should be mentioned in the caption, or just replaced by the actual boundary. Red dot is not mentioned
C2. Eq.(1) sum over \sigma in the first and last term is missing
C3. Eq.(3) 2nd line, +/- in the fist term is missing
C4. Eq.(6) is valid for periodic boundary conditions, while in the appendix we find:
"... and we impose anti-periodic boundary conditions to reduce boundary effects". In this respect I have several questions:
Q1: Which kind of "boundary effects" are avoided by choosing anti-periodic over periodic boundary conditions;
Q2: Is Calabrese-Cardy formula actually valid for the anti-periodic boundary conditions. It is not trivial at all, so either comment or an appropriate reference is necessary here.
Q3: Fig.6: Eq.(6) apart from the central charge contains a non-unversal constant a_0. How exactly the central charge is extracted. If it has been done in a usual way, by fitting the finite-size profile of the entanglement entropy, an example of the fit should be provided.
Q4: How exactly the anti-periodic boundary conditions have been imposed?
Q5: How the anti-periodic boundary conditions affect the value of the dominant wave-vector k in Fig.3,4,5...
Q6: Filling 1/5 is chosen to stabilize the CLL phase. Is there any experimentally relevant motivation for this value?
C5: "...Here we summarize our results in Fig. 6, where we show the infinite size
extrapolation of the central charge". There is no finite-size extrapolation in Fig.6, but it is necessary. In particular, around V=5.7 the central charge decay so fast, so it is hard to believe it will stay at c=5/2 at the thermodynamic limit. Below Fig.6, "(iii) the critical point... is located at Vc... where the central charge jumps to c=5/2". The "jump" is only due to lack of the point on the way to V=5.7.
C6: I find it quite misleading to provide the unphysical results such as c=4 at V=7.4 in Fig.6; or negative gap in Fig.7.
C7: In Fig.6, results for N=60 and 6<V<7 is very different from those of N<60. Is it due to convergence, or there a deep reason for this, e.g. finite-size crossover in the ground-state.
C8: Fig.7 three-parameters fit with only four data points is not very reliable. Behaviour for V=8 is very different from V<7. It would be interesting to see the value of the wave-vector k at each point. If I am not mistaken, it changes with L.
C9: Eq.7: the operator content depends on the boundary conditions, and therefore d_\alpha for periodic boundary condition is not necessary the came as d_\alpha for an anti-periodic one. Comments here would be relevant.
C10: Fig.9, no y-labels
Q7: In appendix: " In particular, we find that keeping tiny blocks in the Eq. (6) leads to
an overestimation of the central charge, signalling the importance of the non-universal effects." First, it has been mentioned in the original paper Ref.[53], that the Eq.6 is only valid for l,L-l>>1. Second, in Fig.13 l<=7 is used in contradiction with the comment above.
Q8: I wonder, whether the authors consider an option of open boundary conditions. Of course, the price to pay is an emergence of the edge effects, but fixing the boundary conditions one can actually profit from the edges and from the boundary conformal field theory. At the same time, in (anti-)periodic system entanglement is associated with two cuts, and therefore is superimposed, that leads to extremely large bond dimension D necessary to reach the convergence. In open boundary conditions there is only one channel of entanglement, so the bond-dimension grows much slower, and one can compute the wave-function for much larger systems sizes.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Report

In the present manuscript, titled “One-dimensional extended Hubbard model with soft-core potential”, authors Thomas Botzung, Guido Pupillo, Pascal Simon, Roberta Citro and Elisa Ercolessi study the ground state properties of a particular Hubbard model with specific long-range interactions termed “soft-shoulder potential”, which extends over several sites without decay, in addition to local repulsion between spin-up and spin-down electrons. The authors present results arguing for the emergence of Cluster Luttinger Liquid (CLL) phases in this model at a particular density and especially for the emergence of a fermionic excitation branch in addition to the usual bosonic ones at the transition from Tomonaga Luttinger liquid to CLL, which they term “supersymmetric”.

In light of substantial recent experimental activity on quantum many-body systems with long-range interactions in neutral dipolar gases, Rydberg atoms, trapped ions and in particular non-decaying long-range interactions mediated via light fields in optical cavities, the work contained in this manuscript deals with an interesting and timely topic. Thus, the authors findings, if substantiated, would certainly be suitable for publication in SciPost and be of interest to the community. However, in its present form the manuscript requires analytical improvements to substantiate key findings before being suitable for publication. These are listed and discussed in the following, along with several minor points on presentation and legibility:

(1) The most interesting single finding in the present manuscript is the authors claim of the emergence of fermionic excitations along with the standard bosonic ones at the transition between TLL and CLL_nn around V_c = 5.73. The findings the authors provide for this however require a more careful analysis than the one currently presented. On pg. 10 the authors state that they supposedly show the infinite size extrapolations of the central charge together with the finite-size data in Fig. 6, but such extrapolations are actually missing from the figure.
Should the authors supply the extrapolations, care should be taken to perform these in a theoretically rigorous manner, based on known analytical results for the system-size dependence of the central charge (there already are several infinite-size extrapolations in the present manuscript that appear to be ad-hoc and not based on known scaling behaviour - see below). This is especially important because the inset in the current Fig. 6 does not necessarily force the reader to the conclusion that c=2.5 is the correct endpoint of convergence. In fact, the size dependence is pronounced when looking at e.g. the final drop from L=50 to L=60, which leads to significant doubts that c=2.5 really is the infinite-size value.
On the same topic, on pg. 12 the authors state that all sound speeds coincide around the transition, but Fig. 9 directly contradicts that statement, with v_s always far below the other two velocities. Further up, above eq. (8), the authors, just assume that K_c = 4 at the transition without any proof, reference or theoretical argument.

Finally, explaining the nature of the supposed supersymmetric point requires improvement, assuming the claim of c=2.5 when 1/L=0 can be substantiated. How is the claim of supersymmetry (every fermion has a boson partner) justified when there are two bosonic modes but only one (Majorana) fermionic one? MPS-based methods, including ITensor, provide a number of ways to manipulate and perform measurements on wave functions - how could these tools be deployed to directly probe the presumed fermionic excitations at the presumed transition point?

(2) The appendix of the manuscript raises a number of critical technical questions requiring further explanations, on which the methodological soundness of this work hinges. Discarded weights or energy variances are not provided for any calculation. But fitting in either one of these quantities is the actually well-defined procedure in MPS-based methods, and NOT in the bond-dimension D. Fitting in D is a pure heuristic without a theoretical basis, that will usually give wrong results when compared against energy-variance or discarded weight extrapolations.
Also, many results are obtained for large U- and V-values (up to 50t), which can lead to very slow convergence requiring many sweeps to allow the particles sufficient opportunity to tunnel past each other - no checks are provided that 20 sweeps are actually enough to accomplish this here.
Then there is the issue of the block lengths used in the fit to eq. (6) (if fitting is actually what is being performed in e.g. Fig. 12 - this is also not clearly explained). On the left of Fig. 12, block lengths up to 15 appear to be included, but on the right only up to 8. And why is the left-hand side showing six different fitting-ranges (upper limits from 3 to 8), instead of just the biggest fitting range, l<=8? What are these “non-universal effects” vaguely invoked for both the supposedly “tiny blocks” (bottom of page 16), but also for blocks with l>=8 (top of page 17)? If these effects are present at l=8, why are they included on the left-hand side of Fig. 12?

(3) In a number of places in the manuscript, purely heuristic fitting procedures are used, seemingly or actually without theoretical support. This is most noticeable in the fits for the spin gap and the single particle gap, Figs. 7 and 8 (left), where a generic power law with three fitting parameters is used to fit 4 or 5 points. As the errors of the initial spin-gaps are seemingly not included in the error of the extrapolation, the chosen fit is almost always delivering a rather small least-square error - unsurprisingly.
Fig. 4 panel (b): 4 points fitted by a+b/L+c/L^2 (from the main text, not mentioned in the caption). If this is motivated by theory, it is not explained.
Fig. 8 panel (b): Is the red dashed line a linear fit of all data points? If not, why?

(4) In the abstract the authors write „[…] similarly to what first predicted in a hard-core bosonic chain in Ref.1“, but in Ref. 1 we found: „[…], we demonstrate that the CLL phase [first predicted in Ref. 37] is separated from a conventional TLL [...]“. Why are the authors referring to Ref. 1 instead of Ref. 37?

(5) The usual (minimalistic) references for DMRG are White 92, 93 and Schollwöck 05 and for MPS (of which ITensor is making extensive use) Schollwöck 11.

(6) Soft-core [six occurences] vs. soft-shoulder [5 occurences]: the authors seem to use these terms interchangeable. This might be confusing, sticking to either one of these formulations may improve legibility.

(7) The authors might consider to add a sketch of the soft-shoulder potential as was provided in Ref. 1 and Ref. 37.

(8) Figure 1 is poorly explained in the current state. As the supposed overview figure of the manuscript, it should provide a comprehensive overview to the reader. It does not do so when the phase abbreviations are not even explained in the caption text, and the position of the supposed supersymmetric transition point is not marked and explained as such - is the red dot, which currently has no explanation at all, meant to denote the transition?

(9) On the bottom of page 4: „Finally, in the appendix, we give some details on the numerics and draw our conclusions“; As the conclusions are given before the appendix, this sentence should be corrected.

(10) Below eq. (1) the various number operators are not defined. And as a side note: index notation is inconsistent, sometimes indicies are separated by a comma and sometimes not.

(11) Check label and axis sizes. E.g., the inset of Fig. 8 panel (b) is extremely small.

(12) Below eq. (3), the definitions of spin-up and spin down numbers are incongruous with the rest of the manuscript: probably the first equals sign should be a plus, leading to N=N_up+N_down?

(13) Right below eq. (4), the definition of the spin-imbalanced ground state energy is potentially confusing. To ensure consistency with eq. (4), it would need to add $\pm 1$ to the subscript.

(14) Typo in 2.1 2) „[…], thus a phase transition form standard TLL to a new [...]“; form -> from.

(15) Last paragraph on page 7: „[…] in the appendix 4.“ should be appendix A.

(16) In 3.2 in the description of Fig. 4 panel (b): The lines are dashed not dotted.

(17) In 3.2 the authors introduce yet another single-particle density n, after having introduced N, N_up, N_down, and <n_up>, <n_down>, and rho. This may lead to confusion on the part of the reader.

(18) In 3.3 „Furthermore, as shown in Fig. 10 panel (c) the double occupancy increases with the interaction strength.“ It should be clarified which interaction parameter exactly is meant here - currently only the figure on the following page allows the reader to deduce that V is meant.

(19) Caption of Fig. 11: „[…] obtained from the Cardy-Calabrese formula […]“. Add a reference to either Eq. (6) or Ref. 53 or both.

(20) ITensor is spelled improperly in the appendix.

(21) In figures throughout the manuscript Hamiltonian parameters are sometimes scaled with t and sometimes not when used as axis labels. Furthermore, some axis labels are missing altogether, e.g., Fig. 7.

• validity: ok
• significance: good
• originality: ok
• clarity: ok
• formatting: reasonable
• grammar: reasonable

### Strengths

1- Numerical study of an extended Hubbard model in one dimension, relevant to some cold atoms experiments.
2- Rich ground-state phase diagram including Tomonaga-Luttinger liquid (TLL) and cluster Luttinger liquid (CLL)

### Weaknesses

1- Most numerical raw data are not presented.
2- The phase diagram is only sketched.
3- The cluster Luttinger liquid (CLL) is not a new phase and has been proposed and found long time ago. References should be included. The additional fermionic mode at the transition has already been presented by some of the authors in a different model. As such the paper lacks novelty to some extent.

### Report

This paper provides mostly a numerical investigation of the ground-state phase diagram of some extended Hubbard model in one dimension. Using standard measurements and DMRG for numerical simulation, a sketch of phase diagram is presented for a fixed density and potential range. In principle, we could expect to have a large part of the phase diagram studied ?

Regarding numerical simulations, the authors have chosen to use antiperiodic boundary conditions, which is known to be hard for DMRG. What is the reason for such a choice ? Apparently this strongly limits the sizes that can be simulated: typically, data are presented only for L=30 or L=50, which is quite small to discuss criticality and provide accurate results. Even though an Appendix does provide some details about simulations, I would like to ask the authors to provide more data, in particular about the entanglement entropy. Indeed the extraction of the central charge seems quite involved and inaccurate, which puts some doubts on the claim of having c=5/2 at the transition, see Fig. 6. In the same figure, it looks extremely strange to have an effective c diverging. I expect that for an SU(2) system with a single charge channel, the maximum of c would be 3+1.

Moreover, I am not very fond of the way that the CLL is presented. In the abstract, it is written that it was first predicted in Ref. [1]. But in the abstract of Ref. [1], it is explicitly written that it was "first predicted" in Ref. [37]... To my knowledge, it is in fact a much older concept, found for instance in multicomponent fermionic chains (Lecheminant et al. PRL 95, 240404 (2005); Roux et al EPJB 68, 293 (2009) etc.) . In fact, even fermionic pairing on a correlated t-J ladder can be seen as a mechanism to go from a Luttinger liquid made of charge-1 objects to a Luther-Emery phase made of charge-2 pairs.

Minor point: the definition of the Luttinger parameter K does not seem to be the standard one (K=4 for SU(2) symmetry) ?

In conclusion, I do not think that the paper can be published in SciPost given its weaknesses, both in terms of physical findings (which are merely incremental) and in terms of data presentation and analysis.

• validity: low
• significance: ok
• originality: low
• clarity: ok
• formatting: good
• grammar: good