SciPost Submission Page
$\eta$‑pairing state in a flat‑band lattice: interband coupling effects on entanglement entropy logarithm
by Seik Pak, Hanbyul Kim, Chan Bin Bark, Sang‑Jin Sin, Jae‑yoon Choi, Moon Jip Park
Submission summary
| Authors (as registered SciPost users): | Seik Pak |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202504_00049v1 (pdf) |
| Date submitted: | April 30, 2025, 2:28 a.m. |
| Submitted by: | Seik Pak |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approach: | Theoretical |
Abstract
The $\eta$-pairing state is the eigenstate of the hypercubic Hubbard model, which exhibits anomalous logarithmic scaling of entanglement entropy. In multi-band systems, $\eta$-pairing can be exact eigenstate when the band is flat without interband coupling. However, typical flatband systems such as Lieb and Kagome lattices often feature band touchings, where interband coupling effects are non-negligible. Using the Creutz ladder, we investigate the deformation of $\eta$-pairing states under the interband coupling effect. Our results show corrections to entanglement entropy scaling, with modified $\eta$-pairing states displaying broadened doublons, nonuniform energy spacing, and deviations from exact behavior for configurations with more than one $\eta$-pair, even in the large band gap limit, except at $t = 0$. Through a Schrieffer--Wolff transformation, we quantify corrections to the spectrum generating algebra, offering insights into the interplay between interaction-driven phenomena and band structure effects. These findings illuminate the robustness and limitations of $\eta$-pairing in realistic flatband systems.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
- This is an interesting extension of the celebrated $\eta$-pairing states to approximate settings.
Weaknesses
- Results not presented clearly, many details missing.
Report
The paper makes an interesting attempt to extend the $\eta$-pairing states to more general settings. However, I think there are many issues that still need to be fixed, and I have listed them in the Requested Changes section.
Requested changes
-
Around Eq.~(17) they claim that the effective Hamiltonian within the Schrieffer-Wolff subspace exhibits $\eta$-pairing to first order. Could the authors provide more details here? What is the effective gap $\Delta$, what is the precise perturbation parameter, and why does it satisfy the $\eta$-pairing condition?
-
Around Eq.~(21) they compute the correction to the $\eta$-pairing spectrum to next order, and say that this remains equally spaced for $U \ll \Delta$. Could they explicitly write out what is $\Delta$ for this model (in terms of $t, t', U$)? It does not seem to have been defined explicitly anywhere.
-
In Fig.~2, they show some data for the energies of the $\eta$-pairing states. The inset seems to suggest that the energies deviate from the analytical prediction, even though the authors claim the contrary in the caption. Please could they clarify this discrepancy?
-
In all of the figures, which of the $\eta$-pairing states are the authors computing? There is an entire tower of states, and for all these plots they need to specify the value of $n$, i.e., which state in the tower are they considering.
-
How do they actually identify the $\eta$-pairing states numerically? In this flat-band $\eta$-pairing settings, is the entire spectrum composed of just the $\eta$-pairing states, or are there other non-$\eta$-pairing states too? If it is the latter, how do they actually find the $\eta$-pairing states numerically? Please could they clarify the procedure?
Recommendation
Ask for minor revision
Strengths
-
Manuscript addresses an interesting subject relevant to the intersection of superconducting many-body states and flat band physics, namely the impact of nearby dispersive bands on paired states in the flat band.
-
Includes an analytical treatment when energetic distance to dispersive band is finite
Weaknesses
-
Presentation is poor in parts: band structure as function of t and t' is not well explained. As a consequence, it is unclear how the band-gap \Delta is controlled by t and t'. Also, the energy shift \epsilon (top of pg. 8; Fig. 2) is neither defined nor explained.
-
Results from many-body numerics are weak and of limited validity (covering at most two pairs on 13 unit cells); the broad statements by the authors for physics at finite band-gap would demand substantially more data.
-
Manuscript appears to contain unproven statement, that the \eta-state is the ground state at negative U (bottom of pg. 9), which would even be incorrect for standard bipartite lattices
Report
As a concrete example, the authors treat the two-leg 1D Creutz-ladder, and aim to show - through a mixture of perturbative analytics and (very) small-scale numerics for one or two pairs - that the \eta-states largely survive the presence of a nearby dispersive bands.
But this statement, as well as some others listed in the "Weaknesses" and "Requested Changes" sections, is is not supported by the data presented in this manuscript - only a systematic scaling analysis, which fixes a density of pairs and systematically extrapolates observables across a range of system sizes can validate the uncontrolled perturbation theory that the authors develop here.
This is regrettable, as for the 1D systems such as the Creutz-ladder studied here there is the very powerful family of matrix product state-based algorithms, as e.g. implemented in the publicly abvailable ITensor or SyTen packages, as the authors will be aware based on their references.
Requested changes
-
Authors need to provide comprehensive numerical evidence for their hypothesis that the tower of eigenstates survives for finite gaps with only U effectively renormalised well beyond the one- and two-pair regime and using a systematic scaling analysis
-
Authors need to substantiate (or discard) claim that the \eta-state is the ground state at negative U for these flat band systems
-
The energy shift \epsilon needs to be well defined and explained
-Relationship between t, t' and \Delta needs to be shown explicitly and ideally illustrated.
-Correct the typo in eq. (13) where t and t' appear only in some terms but not in others.
Recommendation
Ask for major revision
Strengths
Weaknesses
1- The general quality of the paper is rather poor denoting carelessness in its preparation. For instance, there are repeated references (Refs. 15, 21, 22 and Refs. 31, 38), many typos and poor english grammar and the explanations are generally unclear. The acknowledgement section is also repeated twice. Parentheses are missing in Eq. 13. 2- In many case notation and nomenclature are used without defining them first, for instance what is "t=0" in the abstract? What are the authors referring to when they talk about "exact" and "modified" $\eta$-pairing states? Adding references to equations in the text is essential to guide the reader through the results. 3- Fig. 1 is not very useful, while it would be useful to have plots of the band structure of the model under consideration. Also, it would be useful to have at hand some expressions for the band gap and the bandwidth of the upper band as a function of the model parameters. 4- The authors consider a specific instance of a 1D model called the Creutz ladder. Notably the same model has been studied using the SW transformation also in Ref. 27. The model parameters are however different. With the choice of parameter in the present paper, the Wannier functions of the flat band are compact and nonoverlapping. This is known as a trivial flat band, in the sense that it cannot support transport in the presence of interactions. In the case of Ref. 27 instead the compact localized states are overlapping, which is the case in a nontrivial flat band. The calculation using the SW transformation of the corrections due to interband coupling is identical to that of Ref. 27, but in the less interesting case of a trivial flat band, so I do not think that the results in Sec. 2.3 are noteworthing. 5- In Fig. 2a the authors show results for $t=0$. In this case the model reduces to a collection of independent Hubbard dimers. I am not sure why the authors think that it is interesting to study this case. 6- Perhaps there are some interesting results in Sec. 4 and 5, but I fail to grasp even some simple aspects. For instance in Fig. 3, the entanglement entropy and the two-body correlation function are shown in the case of some unspecified "modified $\eta$-pairing" states. The authors should at least explain how these modified states are identified and compare their properties with those of more typical states that obey the volume law for instance. As a consequence, with the amount of information provided in the manuscript, it is impossible to reproduce the numerical results.
Report
Requested changes
There are many things to be done in order to improve the clarity and rigor of the manuscript. Even if this is done, I am not sure if the results presented in the manuscript would eventually be enough for the publication on SciPost, given that they are very limited in scope and have a substantial overlap with at least one previous work. A definite judgement can be given only after the clarity of the manuscript has been improve, especially regarding Sec. 4 and 5.
Recommendation
Reject