arxiv: 2512.09901 · v1 · submitted 2025-12-10 · ❄️ cond-mat.supr-con · cond-mat.mtrl-sci

Superconductivity and geometric superfluid weight of a tunable flat band system

M. A. Mojarro , Sergio E. Ulloa This is my paper

Pith reviewed 2026-05-16 23:03 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.mtrl-sci

keywords superconductivityflat bandssuperfluid weightquantum metricalpha-T3 latticeHubbard modelBKT transition

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The pith

Quasi-flat bands in the tunable α-T3 lattice produce a power-law superconducting gap and linearly growing geometric superfluid weight.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates superconductivity in a two-dimensional α-T3 lattice that hosts an isolated quasi-flat band whose bandwidth is controlled by a single parameter α. Within a mean-field treatment of the attractive Hubbard model, the superconducting gap at the quasi-flat filling rises as a power law with increasing interaction strength, rather than the conventional exponential form, because the density of states diverges. The superfluid weight splits into conventional and geometric pieces; the geometric piece, set by the quantum metric, grows linearly at weak coupling and is further strengthened by raising α, while on-site asymmetries broaden the band and boost the conventional piece. The Berezinskii-Kosterlitz-Thouless temperature follows the same enhancement with α.

Core claim

In the mean-field solution of the attractive Hubbard model on the α-T3 lattice, the superconducting order parameters on the three sublattices depend strongly on α, interaction strength, and filling. At quasi-flat band filling the gap opens and grows as a power law with interaction strength due to the diverging density of states. Linear-response calculation of the superfluid weight shows that the conventional (band-derivative) term is suppressed while the geometric term, dominated by the quantum metric, grows linearly at small interaction and is enhanced by tuning α, especially near half-filling. The resulting Berezinskii-Kosterlitz-Thouless temperature increases markedly with α.

What carries the argument

The quantum metric of the isolated quasi-flat band, which supplies the geometric contribution to the superfluid weight once band dispersion is suppressed.

Load-bearing premise

The mean-field approximation of the attractive Hubbard model remains valid and captures the dominant physics of the superconducting state and superfluid weight in the quasi-flat band regime.

What would settle it

Measuring the superconducting gap size as a function of interaction strength at the quasi-flat filling and testing whether the dependence is power-law rather than exponential.

Figures

Figures reproduced from arXiv: 2512.09901 by M. A. Mojarro, Sergio E. Ulloa.

**Figure 3.** Figure 3: FIG. 3. Bandwidth of the quasi-flat band [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. (a) Zero temperature superconducting order param [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Superfluid weight as a function of [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. (a) Order parameters and (b) conventional and ge [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Conventional and geometric superfluid weight at half [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. (a) Surface plot of the quantum metric of the flat [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. (a) Superfluid weight at half-filling as a function [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗

read the original abstract

We study superconductivity and superfluid weight of the two-dimensional $\alpha$-$\mathcal{T}_3$ lattice with on-site asymmetries, hosting an isolated quasi-flat band with tunable bandwidth via a parameter $\alpha$. Within a mean-field approximation of the attractive Hubbard model, we obtain the superconducting order parameters on the three inequivalent sublattices and show their strong dependence on $\alpha$, interaction strength, and electron filling. At quasi-flat band filling, a superconducting gap opens and grows power-law fast with interaction strength, instead of the usual slow exponential growth, due to diverging density of states. We calculate the superfluid weight from linear response theory and study its band dispersion and geometric contributions. While the conventional part proportional to band derivatives is suppressed in the quasi-flat band regime, the contribution dominated by the quantum metric grows linearly for small interaction strength. We further demonstrate how tuning $\alpha$ enhances the quantum metric and thus the geometric superfluid weight especially near half-filling, while increasing on-site asymmetries increases the conventional contribution by broadening the quasi-flat band. We obtain the Berezinskii-Kosterlitz-Thouless transition temperature and demonstrate its strong dependence and enhancement with the parameter $\alpha$. Our results establish a tunable flat band system, the $\alpha$-$\mathcal{T}_3$ lattice model, as a candidate for tunable quantum geometry and superfluid weight and as a prototype of related behavior in tunable quantum materials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a tunable α-T3 model that separates geometric and conventional superfluid weight contributions under mean-field, but the flat-band regime puts the approximation on shaky ground.

read the letter

The core result is a concrete lattice where α tunes both the bandwidth of an isolated quasi-flat band and the quantum metric. In mean-field treatment of the attractive Hubbard model they find that at the quasi-flat filling the gap opens as a power law in U rather than exponentially, and the geometric piece of the superfluid weight grows linearly with small U while the conventional piece stays suppressed. Tuning α toward the flat limit boosts the geometric weight, especially near half filling, and they also track the BKT temperature. That combination of tunable asymmetry, explicit separation of the two superfluid-weight channels, and the reported scalings is not standard in the existing flat-band literature. The calculations are straightforward linear-response plus mean-field, and the model itself is clean enough to be useful for thinking about geometry-driven pairing. The main limitation is that the headline claims rest entirely on the saddle-point approximation. Once α makes the bandwidth arbitrarily small the density of states diverges, so modest U already takes the system outside the weak-coupling window where mean-field is controlled. No quantitative bound on U/W(α) or comparison to fluctuation-corrected or exact methods is given, so the power-law gap and linear geometric weight could shift once corrections are included. The paper is aimed at the flat-band superconductivity and quantum-geometry community. It is worth sending to referees because the model is new and the separation of contributions is worth checking, even if the mean-field results need a validity window and possibly some beyond-mean-field tests before they can be taken as quantitative.

Referee Report

2 major / 2 minor

Summary. The manuscript studies superconductivity and superfluid weight in the two-dimensional α-T₃ lattice with tunable quasi-flat band via on-site asymmetries. Within mean-field theory applied to the attractive Hubbard model, it reports that at quasi-flat band filling the superconducting gap opens with power-law dependence on interaction strength U (instead of exponential) due to diverging DOS; the geometric contribution to the superfluid weight (from quantum metric) grows linearly with small U and is enhanced by tuning α, while the conventional term is suppressed; the BKT temperature is computed and shown to depend strongly on α.

Significance. If the mean-field results are robust, the work identifies the α-T₃ lattice as a tunable platform for isolating geometric contributions to superfluidity in flat-band systems and for engineering enhanced BKT temperatures, with potential relevance to quantum materials hosting engineered flat bands.

major comments (2)

[mean-field gap equation and linear-response superfluid weight sections] The headline power-law gap opening Δ ∝ U^β and the linear-in-U geometric superfluid weight are obtained entirely from the mean-field gap equation and linear-response formula applied to the mean-field Hamiltonian. No quantitative window of validity (e.g., U/W(α) ≪ 1) or comparison to fluctuation-corrected or numerically exact methods is provided, even though tuning α drives W(α) → 0 and places the system outside the controlled weak-coupling regime; this directly undermines the claimed scalings.
[superfluid weight decomposition] The geometric superfluid weight is extracted from the quantum metric of the Bloch states of the mean-field Hamiltonian; because the order parameter Δ(U,α) itself is mean-field, any correction to Δ propagates into the reported linear scaling and α-enhancement. The manuscript does not test the sensitivity of these quantities to beyond-mean-field corrections.

minor comments (2)

[model Hamiltonian] Clarify the precise definition of the on-site asymmetry parameters and their relation to the three sublattice order parameters in the gap equations.
[numerical methods] Add explicit error estimates or convergence checks for the numerical solution of the gap equations at high DOS.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and have incorporated partial revisions to clarify the scope and limitations of the mean-field approach.

read point-by-point responses

Referee: [mean-field gap equation and linear-response superfluid weight sections] The headline power-law gap opening Δ ∝ U^β and the linear-in-U geometric superfluid weight are obtained entirely from the mean-field gap equation and linear-response formula applied to the mean-field Hamiltonian. No quantitative window of validity (e.g., U/W(α) ≪ 1) or comparison to fluctuation-corrected or numerically exact methods is provided, even though tuning α drives W(α) → 0 and places the system outside the controlled weak-coupling regime; this directly undermines the claimed scalings.

Authors: We agree that the reported scalings are obtained within mean-field theory applied to the attractive Hubbard model. The power-law gap opening follows directly from the divergent density of states at quasi-flat-band filling in the gap equation, replacing the usual BCS exponential form. The linear-in-U geometric superfluid weight arises in the small-Δ limit of the linear-response formula, where the leading term is set by the quantum metric of the underlying bands. We acknowledge that tuning α reduces W(α) and eventually takes the system outside the strict weak-coupling regime. In the revised manuscript we have added a dedicated paragraph in the discussion section that (i) estimates the regime of validity as U/W(α) ≲ 1 for finite α, (ii) notes that the qualitative distinction from exponential BCS behavior remains robust for moderate α, and (iii) references existing literature on mean-field treatments of flat-band superconductivity. We have not performed beyond-mean-field calculations, as these lie outside the present scope. revision: partial
Referee: [superfluid weight decomposition] The geometric superfluid weight is extracted from the quantum metric of the Bloch states of the mean-field Hamiltonian; because the order parameter Δ(U,α) itself is mean-field, any correction to Δ propagates into the reported linear scaling and α-enhancement. The manuscript does not test the sensitivity of these quantities to beyond-mean-field corrections.

Authors: The quantum metric is evaluated on the self-consistent mean-field bands, so corrections to Δ do in principle affect the result. However, for small U the geometric contribution is dominated by the non-interacting quantum metric, with interaction-induced corrections entering only at O(Δ²). Because the gap equation yields Δ ∝ U^β with β > 0, the leading U-linear behavior is preserved. In the revised manuscript we have added supplementary calculations in which Δ is varied independently around the self-consistent value; these confirm that the linear scaling and α-enhancement remain intact within the small-U window. A systematic beyond-mean-field assessment would require methods such as DMFT or quantum Monte Carlo on the α-T₃ lattice, which are computationally demanding and beyond the scope of this work. revision: partial

standing simulated objections not resolved

Providing quantitative comparisons of the mean-field gap and superfluid weight to fluctuation-corrected or numerically exact methods for the α-T₃ model.

Circularity Check

0 steps flagged

No circularity: results follow from direct mean-field solution and standard linear-response formulas

full rationale

The derivation begins from the attractive Hubbard Hamiltonian on the α-T3 lattice, applies the standard mean-field decoupling to obtain self-consistent order parameters, and evaluates the superfluid weight via the usual linear-response expression that separates conventional (band-velocity) and geometric (quantum-metric) pieces. The reported power-law gap opening at quasi-flat filling is a direct numerical consequence of the divergent DOS already present in the non-interacting band structure; the linear-in-U growth of the geometric weight likewise follows from the small-U expansion of the same response formula applied to the mean-field state. No parameter is fitted to a target observable and then re-labeled as a prediction, no uniqueness theorem is imported from self-citation, and no ansatz is smuggled via prior work. The entire chain is therefore self-contained within the model and the chosen approximation.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claims rest on the mean-field decoupling of the attractive Hubbard interaction and on the validity of linear response for the superfluid weight; no new particles or forces are introduced.

free parameters (3)

α
Lattice asymmetry parameter that tunes the flat-band bandwidth; its value is chosen to control the regime of interest.
U
On-site attractive interaction strength; results are reported as functions of U.
filling
Electron density, especially at quasi-flat-band filling.

axioms (2)

domain assumption Mean-field approximation of the attractive Hubbard model is sufficient to describe the superconducting state.
Invoked throughout the calculation of order parameters and superfluid weight.
standard math Linear response theory yields the superfluid weight from the current-current correlation function.
Standard in condensed-matter theory for superfluid stiffness.

pith-pipeline@v0.9.0 · 5562 in / 1440 out tokens · 52078 ms · 2026-05-16T23:03:06.720140+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Within a mean-field approximation of the attractive Hubbard model, we obtain the superconducting order parameters... At quasi-flat band filling, a superconducting gap opens and grows power-law fast with interaction strength... We calculate the superfluid weight from linear response theory and study its band dispersion and geometric contributions.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the geometric contribution dominated by the quantum metric grows linearly for small interaction strength... tuning α enhances the quantum metric

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages · 1 internal anchor

[1]

Liu, X.-B

T. Liu, X.-B. Qiang, H.-Z. Lu, and X. C. Xie, Quantum geometry in condensed matter, National Science Review 12, nwae334 (2024)

work page 2024
[2]

J. Yu, B. A. Bernevig, R. Queiroz, E. Rossi, P. T¨ orm¨ a, and B.-J. Yang, Quantum geometry in quantum materi- als, npj Quantum Materials10, 101 (2025)

work page 2025
[3]

A. Gao, N. Nagaosa, N. Ni, and S.-Y. Xu, Quantum Geometry Phenomena in Condensed Matter Systems (2025), arXiv:2508.00469 [cond-mat.str-el]

work page arXiv 2025
[4]

M. Kang, S. Kim, Y. Qian, P. M. Neves, L. Ye, J. Jung, D. Puntel, F. Mazzola, S. Fang, C. Jozwiak, A. Bost- wick, E. Rotenberg, J. Fuji, I. Vobornik, J.-H. Park, J. G. 8 Checkelsky, B.-J. Yang, and R. Comin, Measurements of the quantum geometric tensor in solids, Nature Physics 21, 110 (2025)

work page 2025
[5]

J. P. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Communications in Math- ematical Physics76, 289 (1980)

work page 1980
[6]

Resta, The insulating state of matter: a geometrical theory, The European Physical Journal B79, 121 (2011)

R. Resta, The insulating state of matter: a geometrical theory, The European Physical Journal B79, 121 (2011)

work page 2011
[7]

M. V. Berry, Quantal phase factors accompanying adia- batic changes, Proceedings of the Royal Society of Lon- don. A. Mathematical and Physical Sciences392, 45 (1984)

work page 1984
[8]

M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys.82, 3045 (2010)

work page 2010
[9]

Qi and S.-C

X.-L. Qi and S.-C. Zhang, Topological insulators and su- perconductors, Rev. Mod. Phys.83, 1057 (2011)

work page 2011
[10]

T¨ orm¨ a, Essay: Where Can Quantum Geometry Lead Us?, Phys

P. T¨ orm¨ a, Essay: Where Can Quantum Geometry Lead Us?, Phys. Rev. Lett.131, 240001 (2023)

work page 2023
[11]

Verma and R

N. Verma and R. Queiroz, Instantaneous response and quantum geometry of insulators, Proceedings of the Na- tional Academy of Sciences122, e2405837122 (2025)

work page 2025
[12]

Ahn, G.-Y

J. Ahn, G.-Y. Guo, N. Nagaosa, and A. Vishwanath, Rie- mannian geometry of resonant optical responses, Nature Physics18, 290 (2022)

work page 2022
[13]

Gao, Y.-F

A. Gao, Y.-F. Liu, J.-X. Qiu, B. Ghosh, T. V. Trevisan, Y. Onishi, C. Hu, T. Qian, H.-J. Tien, S.-W. Chen, M. Huang, D. B´ erub´ e, H. Li, C. Tzschaschel, T. Dinh, Z. Sun, S.-C. Ho, S.-W. Lien, B. Singh, K. Watanabe, T. Taniguchi, D. C. Bell, H. Lin, T.-R. Chang, C. R. Du, A. Bansil, L. Fu, N. Ni, P. P. Orth, Q. Ma, and S.-Y. Xu, Quantum metric nonlinear ...

work page 2023
[14]

N. Wang, D. Kaplan, Z. Zhang, T. Holder, N. Cao, A. Wang, X. Zhou, F. Zhou, Z. Jiang, C. Zhang, S. Ru, H. Cai, K. Watanabe, T. Taniguchi, B. Yan, and W. Gao, Quantum-metric-induced nonlinear transport in a topo- logical antiferromagnet, Nature621, 487 (2023)

work page 2023
[15]

Komissarov, T

I. Komissarov, T. Holder, and R. Queiroz, The quan- tum geometric origin of capacitance in insulators, Nature Communications15, 4621 (2024)

work page 2024
[16]

Verma, P

N. Verma, P. J. W. Moll, T. Holder, and R. Queiroz, Quantum Geometry: Revisiting electronic scales in quan- tum matter (2025), arXiv:2504.07173 [cond-mat.mtrl- sci]

work page arXiv 2025
[17]

Fern´ andez-M´ endez, R

Y. Fern´ andez-M´ endez, R. Carrillo-Bastos, and J. A. May- torena, Probing quantum geometric capacitance via pho- tonic spin Hall effect, Phys. Rev. B112, L121407 (2025)

work page 2025
[18]

Peotta and P

S. Peotta and P. T¨ orm¨ a, Superfluidity in topologically nontrivial flat bands, Nature Communications6, 8944 (2015)

work page 2015
[19]

Liang, T

L. Liang, T. I. Vanhala, S. Peotta, T. Siro, A. Harju, and P. T¨ orm¨ a, Band geometry, Berry curvature, and super- fluid weight, Phys. Rev. B95, 024515 (2017)

work page 2017
[20]

Peotta, K.-E

S. Peotta, K.-E. Huhtinen, and P. T¨ orm¨ a, Quantum ge- ometry in superfluidity and superconductivity (2023), arXiv:2308.08248 [cond-mat.quant-gas]

work page arXiv 2023
[21]

X. Hu, T. Hyart, D. I. Pikulin, and E. Rossi, Geo- metric and Conventional Contribution to the Superfluid Weight in Twisted Bilayer Graphene, Phys. Rev. Lett. 123, 237002 (2019)

work page 2019
[22]

Julku, T

A. Julku, T. J. Peltonen, L. Liang, T. T. Heikkil¨ a, and P. T¨ orm¨ a, Superfluid weight and Berezinskii- Kosterlitz-Thouless transition temperature of twisted bi- layer graphene, Phys. Rev. B101, 060505 (2020)

work page 2020
[23]

Tanaka, J

M. Tanaka, J. ˆI.-j. Wang, T. H. Dinh, D. Rodan-Legrain, S. Zaman, M. Hays, A. Almanakly, B. Kannan, D. K. Kim, B. M. Niedzielski, K. Serniak, M. E. Schwartz, K. Watanabe, T. Taniguchi, T. P. Orlando, S. Gus- tavsson, J. A. Grover, P. Jarillo-Herrero, and W. D. Oliver, Superfluid stiffness of magic-angle twisted bilayer graphene, Nature638, 99 (2025)

work page 2025
[24]

Julku, S

A. Julku, S. Peotta, T. I. Vanhala, D.-H. Kim, and P. T¨ orm¨ a, Geometric Origin of Superfluidity in the Lieb- Lattice Flat Band, Phys. Rev. Lett.117, 045303 (2016)

work page 2016
[25]

R. P. S. Penttil¨ a, K.-E. Huhtinen, and P. T¨ orm¨ a, Flat- band ratio and quantum metric in the superconductivity of modified Lieb lattices, Communications Physics8, 50 (2025)

work page 2025
[26]

Wu, X.-F

Y.-R. Wu, X.-F. Zhang, C.-F. Liu, W.-M. Liu, and Y.- C. Zhang, Superfluid density and collective modes of fermion superfluid in dice lattice, Scientific Reports11, 13572 (2021)

work page 2021
[27]

Thumin and G

M. Thumin and G. Bouzerar, Flat-band superconductiv- ity in a system with a tunable quantum metric: The stub lattice, Phys. Rev. B107, 214508 (2023)

work page 2023
[28]

Thumin and G

M. Thumin and G. Bouzerar, Crossing over from flat band superconductivity to conventional superconductiv- ity (2025), arXiv:2507.07701 [cond-mat.supr-con]

work page arXiv 2025
[29]

Raoux, M

A. Raoux, M. Morigi, J.-N. Fuchs, F. Pi´ echon, and G. Montambaux, From Dia- to Paramagnetic Orbital Susceptibility of Massless Fermions, Phys. Rev. Lett. 112, 026402 (2014)

work page 2014
[30]

Illes, J

E. Illes, J. P. Carbotte, and E. J. Nicol, Hall quanti- zation and optical conductivity evolution with variable Berry phase in theα−T 3 model, Phys. Rev. B92, 245410 (2015)

work page 2015
[31]

M. A. Mojarro, V. G. Ibarra-Sierra, J. C. Sandoval- Santana, R. Carrillo-Bastos, and G. G. Naumis, Electron transitions for Dirac Hamiltonians with flat bands un- der electromagnetic radiation: Application to theα− ⊔ 3 graphene model, Phys. Rev. B101, 165305 (2020)

work page 2020
[32]

Dey and T

B. Dey and T. K. Ghosh, Floquet topological phase tran- sition in theα−T 3 lattice, Phys. Rev. B99, 205429 (2019)

work page 2019
[33]

Illes and E

E. Illes and E. J. Nicol, Klein tunneling in theα−T 3 model, Phys. Rev. B95, 235432 (2017)

work page 2017
[34]

Illes and E

E. Illes and E. J. Nicol, Magnetic properties of theα−T 3 model: Magneto-optical conductivity and the Hofstadter butterfly, Phys. Rev. B94, 125435 (2016)

work page 2016
[35]

Dey and T

B. Dey and T. K. Ghosh, Photoinduced valley and electron-hole symmetry breaking inα−T 3 lattice: The role of a variable Berry phase, Phys. Rev. B98, 075422 (2018)

work page 2018
[36]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys.81, 109 (2009)

work page 2009
[37]

S. Geng, X. Wang, R. Guo, C. Qiu, F. Chen, Q. Wang, K. Li, P. Hao, H. Liang, Y. Huang, Y. Wu, S. Cui, Z. Sun, T. K. Kim, C. Cacho, D. S. Dessau, B. T. Zhou, and H. Li, Experimental realization of dice-lattice flat band at the Fermi level in layered electride YCl (2025), arXiv:2508.21311 [cond-mat.str-el]

work page internal anchor Pith review Pith/arXiv arXiv 2025
[38]

Pi´ echon, J.-N

F. Pi´ echon, J.-N. Fuchs, A. Raoux, and G. Mon- tambaux, Tunable orbital susceptibility inα−T 3 tight- binding models, Journal of Physics: Conference Series 603, 012001 (2015). 9

work page 2015
[39]

Leykam, A

D. Leykam, A. Andreanov, and S. Flach, Artificial flat band systems: from lattice models to experiments, Ad- vances in Physics: X3, 1473052 (2018)

work page 2018
[40]

Zhao and A

E. Zhao and A. Paramekanti, BCS-BEC Crossover on the Two-Dimensional Honeycomb Lattice, Phys. Rev. Lett. 97, 230404 (2006)

work page 2006
[41]

N. B. Kopnin, T. T. Heikkil¨ a, and G. E. Volovik, High-temperature surface superconductivity in topologi- cal flat-band systems, Phys. Rev. B83, 220503 (2011)

work page 2011
[42]

G. D. Mahan,Many-particle physics(Springer Science & Business Media, 2013)

work page 2013
[43]

D. J. Scalapino, S. R. White, and S. Zhang, Insulator, metal, or superconductor: The criteria, Phys. Rev. B47, 7995 (1993)

work page 1993
[44]

J. M. Kosterlitz and D. J. Thouless, Ordering, metasta- bility and phase transitions in two-dimensional systems, Journal of Physics C: Solid State Physics6, 1181 (1973)

work page 1973
[45]

D. R. Nelson and J. M. Kosterlitz, Universal jump in the superfluid density of two-dimensional superfluids, Phys. Rev. Lett.39, 1201 (1977)

work page 1977