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arxiv: 2506.01764 · v4 · submitted 2025-06-02 · ❄️ cond-mat.supr-con · cond-mat.str-el

Unconventional Superconductivity in mathrm{La₃Ni₂O₇} from the Perspective of Symmetry

Guan-Hao Feng , Jun Quan , Yusheng Hou This is my paper

Pith reviewed 2026-05-19 11:53 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.str-el
keywords La3Ni2O7nickelate superconductivitys±-wave pairingtwo-gap superconductivityorbital pairingthin filmsymmetry analysisDFT+U
0
0 comments X

The pith

La₃Ni₂O₇ shows s±-wave two-gap superconductivity in both forms, but bulk relies on out-of-plane dz² pairing while thin films use in-plane dx²-y² pairing.

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

The paper introduces a symmetry-based phenomenological method to analyze the gap structure of La₃Ni₂O₇. Combining DFT+U results with measured transition temperatures and crystal symmetry, it concludes that both pressurized bulk samples and ambient-pressure thin films display s±-wave pairing symmetry together with two distinct gaps. The microscopic drivers differ sharply: bulk superconductivity is dominated by out-of-plane pairing of Ni-dz² orbitals between layers, whereas thin-film superconductivity is carried mainly by in-plane pairing of Ni-dx²-y² orbitals. The drop in Tc to roughly half its bulk value is traced to this orbital switch, which follows from the reduced ratio of interlayer to intralayer hoppings once the material is made into a film. The approach underscores how symmetry and orbital character together control the pairing mechanism in this layered nickelate.

Core claim

Both pressurized bulk and thin-film La₃Ni₂O₇ exhibit s±-wave pairing symmetry and two-gap superconductivity, yet their dominant microscopic pairing configurations are distinct: superconductivity in the pressurized bulk is dominated by the out-of-plane pairing of the Ni-dz² orbitals, while in the thin film the in-plane pairing of the Ni-dx²-y² orbitals prevails. The observed reduction in Tc is attributed to this transition of the dominant pairing type, driven by the decreased ratio of inter-layer to intra-layer hoppings in the thin film.

What carries the argument

A phenomenological symmetry-based method that combines DFT+U electronic-structure calculations with experimental Tc values and the known crystal symmetry to identify the dominant orbital pairing channels and overall gap symmetry.

If this is right

  • Both the bulk and thin-film forms realize s±-wave symmetry with two gaps.
  • Bulk pairing is carried primarily by out-of-plane Ni-dz² orbitals.
  • Thin-film pairing is carried primarily by in-plane Ni-dx²-y² orbitals.
  • The Tc reduction follows from the shift in dominant pairing when the interlayer hopping ratio falls.
  • The symmetry-based method can be applied to other unconventional superconductors.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Strain or layer-spacing engineering could be used to select which orbital channel dominates and thereby raise Tc in thin films.
  • Similar orbital-selectivity arguments may help interpret pairing in other layered nickelates or cuprates where interlayer coupling varies.
  • Direct probes such as ARPES on both sample types would provide an independent test of the predicted orbital switch.

Load-bearing premise

DFT+U calculations plus measured Tc and structural symmetry are enough to uniquely identify which orbital pairing dominates and to tie the Tc drop directly to that orbital change, without needing direct experimental maps of the gap.

What would settle it

Angle-resolved photoemission or tunneling spectra that show the same dominant orbital character and gap symmetry in both the pressurized bulk and the thin film, or that find no correlation between Tc and the interlayer-to-intralayer hopping ratio.

Figures

Figures reproduced from arXiv: 2506.01764 by Guan-Hao Feng, Jun Quan, Yusheng Hou.

Figure 1
Figure 1. Figure 1: (a) Band structure of the bilayer tight-binding [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Pairing strength ∆i vs Vi obtained by solving the self-consistent BCS gap equations Eq. 6 in the A1g pairing channel. (b) BdG condensation energy FBdG vs |∆i| at Vi = 0.5 eV. The result verifies that the values of |∆i| in (a) are do the saddle points of the corresponding BdG condensation energy. The comparison shows that the s±-wave pairings are dominant, while the dx2−y2 -wave pairings are subdominant… view at source ↗
Figure 3
Figure 3. Figure 3: (a)-(d) Nodal bulk spectra of the superconducting states in the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

The recently discovered superconductor $\mathrm{La_{3}Ni_{2}O_{7}}$ has attracted significant attention due to its remarkably high transition temperature ($T_{c}$) under high pressure. Shortly after this discovery, thin-film $\mathrm{La_{3}Ni_{2}O_{7}}$ was demonstrated to exhibit ambient-pressure superconductivity; however, the corresponding $T_c$ is only about half that of the pressurized bulk material. This striking difference raises questions about the underlying mechanisms governing superconductivity in these two structures. To address this issue, we develop a phenomenological symmetry-based method to investigate the superconducting gap structure in $\mathrm{La_{3}Ni_{2}O_{7}}$. Using density-functional theory methods (DFT+$U$), together with the experimentally determined $T_c$ and structural symmetry, we find that both pressurized bulk and thin-film $\mathrm{La_{3}Ni_{2}O_{7}}$ exhibit $s_{\pm}$-wave pairing symmetry and two-gap superconductivity, yet their dominant microscopic pairing configurations are distinct. In the pressurized bulk, superconductivity is dominated by the out-of-plane pairing of the Ni-$d_{z^2}$ orbitals, while in the thin film, the in-plane pairing of the Ni-$d_{x^2-y^2}$ orbitals prevails. Furthermore, the observed reduction in $T_c$ can be attributed to this transition of the dominant pairing type, driven by the decreased ratio of inter-layer to intra-layer hoppings in the thin film. Our result sheds lights on the microscopic pairing in $\mathrm{La_{3}Ni_{2}O_{7}}$ and reveals the significance of the symmetry. This method can potentially be generalized to a broader range of unconventional superconductors.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a phenomenological symmetry-based method combining DFT+U calculations, the experimentally measured Tc, and crystal symmetry to determine the superconducting gap structure in La3Ni2O7. It concludes that both pressurized bulk and thin-film samples show s±-wave pairing symmetry with two-gap superconductivity, but with distinct dominant microscopic pairing channels: out-of-plane Ni-dz² orbital pairing in the bulk versus in-plane Ni-dx²-y² orbital pairing in the thin film. The lower Tc in the thin film is attributed to this shift in dominant pairing, caused by the reduced inter-layer to intra-layer hopping ratio.

Significance. If the orbital dominance assignments prove robust and non-circular, the work would clarify orbital-selective pairing in nickelate superconductors and provide a symmetry-based explanation for the Tc difference between bulk and thin-film La3Ni2O7. The suggested generalization to other unconventional superconductors could be useful if the method's uniqueness is established.

major comments (2)
  1. [Phenomenological method] Phenomenological method (abstract and main text): The procedure uses the experimentally determined Tc as an input together with DFT+U hoppings and symmetry to select the dominant pairing channel. This creates a circularity risk when the same Tc reduction is then attributed to the resulting transition in orbital dominance, since the quantity being explained participates in fixing the pairing assignment.
  2. [Dominant pairing assignment] Dominant pairing assignment (results section): The claim that DFT+U hoppings plus measured Tc and symmetry uniquely fix out-of-plane dz² dominance for bulk versus in-plane dx²-y² dominance for the thin film is not established. Because the pairing interaction vertex is not computed independently, multiple orbital-selective interaction strengths can reproduce the same s± two-gap spectrum and numerical Tc for the given hoppings, rendering the dominance assignment and its causal link to the hopping ratio non-unique.
minor comments (2)
  1. [Abstract] Abstract: The statement that 'DFT+U plus Tc and symmetry yield the pairing assignments' provides no details on model construction, specific Hubbard U values, or validation against independent gap-structure data.
  2. [Notation] Notation: Ensure uniform typesetting of s± and orbital labels (d_{z^2}, d_{x^2-y^2}) across equations and figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below. Where the concerns identify areas for improved clarity, we will revise the text accordingly while preserving the core phenomenological approach.

read point-by-point responses

  1. Referee: [Phenomenological method] Phenomenological method (abstract and main text): The procedure uses the experimentally determined Tc as an input together with DFT+U hoppings and symmetry to select the dominant pairing channel. This creates a circularity risk when the same Tc reduction is then attributed to the resulting transition in orbital dominance, since the quantity being explained participates in fixing the pairing assignment.

    Authors: We acknowledge the referee's concern about potential circularity. The method assigns the dominant channel independently for each system: for the pressurized bulk, the DFT+U hoppings (with their specific inter-layer component) together with symmetry and the measured higher Tc select out-of-plane dz2 pairing; for the thin film, the same procedure with its reduced inter-layer to intra-layer hopping ratio and measured lower Tc selects in-plane dx2-y2 pairing. The Tc difference is then interpreted as a consequence of the structural change in hopping that switches which channel dominates. To eliminate any ambiguity, we will add a dedicated paragraph in the methods section that explicitly separates the per-system assignment step from the subsequent comparison of the two systems. revision: partial

  2. Referee: [Dominant pairing assignment] Dominant pairing assignment (results section): The claim that DFT+U hoppings plus measured Tc and symmetry uniquely fix out-of-plane dz² dominance for bulk versus in-plane dx²-y² dominance for the thin film is not established. Because the pairing interaction vertex is not computed independently, multiple orbital-selective interaction strengths can reproduce the same s± two-gap spectrum and numerical Tc for the given hoppings, rendering the dominance assignment and its causal link to the hopping ratio non-unique.

    Authors: We agree that the analysis is phenomenological and does not derive the pairing vertex from first principles; therefore the assignment cannot be claimed to be unique in an ab initio sense. Within the symmetry-constrained space and given the orbital-projected hoppings, the measured Tc serves to identify the dominant channel that reproduces both the s± two-gap structure and the numerical value of Tc. We will revise the results and discussion sections to state this limitation explicitly, remove any implication of uniqueness, and note that alternative orbital-selective interaction strengths could in principle yield similar spectra, while still presenting the symmetry-based inference as the most consistent interpretation of the available data. revision: yes

Circularity Check

1 steps flagged

Tc used as input to assign dominant orbital pairing channels; observed Tc reduction then attributed to the resulting channel switch

specific steps
  1. fitted input called prediction [Abstract]
    "Using density-functional theory methods (DFT+U), together with the experimentally determined Tc and structural symmetry, we find that both pressurized bulk and thin-film La3Ni2O7 exhibit s±-wave pairing symmetry and two-gap superconductivity, yet their dominant microscopic pairing configurations are distinct. [...] the observed reduction in Tc can be attributed to this transition of the dominant pairing type, driven by the decreased ratio of inter-layer to intra-layer hoppings in the thin film."

    The dominant pairing configurations (out-of-plane d_z2 for bulk vs. in-plane d_x2-y2 for film) are determined by feeding the measured Tc values into the symmetry-based construction. The reduction in Tc is then explained as a consequence of the transition between those same configurations. The explanatory step therefore reduces to re-describing the input Tc difference rather than deriving the Tc change independently from the hopping ratio.

full rationale

The paper's phenomenological symmetry-based method incorporates the experimentally measured Tc (along with DFT+U hoppings and crystal symmetry) to identify which orbital channel dominates in each structure. The central explanatory claim then attributes the lower Tc in the thin film to the shift in dominance that was itself selected to be consistent with the measured Tc values. This creates a partial circularity in the causal attribution even though the underlying symmetry classification and hopping calculations retain independent content from the DFT+U band structure.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard DFT+U methodology and symmetry principles from condensed-matter theory, with Tc used as an input constraint; no new entities are postulated.

free parameters (1)
  • Hubbard U parameter
    Common adjustable parameter in DFT+U calculations for transition-metal oxides; value not specified in abstract but required to match electronic structure.
axioms (1)
  • domain assumption Superconducting gap structure and dominant orbital pairing can be inferred from crystal symmetry, structural parameters, and measured Tc using a phenomenological model.
    This is the core premise of the symmetry-based method described in the abstract.

pith-pipeline@v0.9.0 · 5863 in / 1513 out tokens · 46566 ms · 2026-05-19T11:53:57.230859+00:00 · methodology

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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
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    we systematically examine all symmetry-allowed multi-orbital superconducting pairings at the Bogoliubov-de Gennes (BdG) mean-field level... By solving the self-consistent gap equations and analyzing the BdG condensation energies, we find that the A1g pairing channel is the most probable one. The dominant pairing is s±-wave, originating from the intra-orbital interaction of the bilayer Ni-d3z2−r2 orbital

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    we employ theory of SI [49–58] to investigate the topological features induced by the possible unconventional pairing symmetries... the refined SI group is XBdG_BS = Z2 × Z2

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Reference graph

Works this paper leans on

73 extracted references · 73 canonical work pages

  1. [1]

    Y. Meng, Y. Yang, H. Sun, S. Zhang, J. Luo, L. Chen, X. Ma, M. Wang, F. Hong, X. Wang, and X. Yu, Density- wave-like gap evolution inLa3Ni2O7 under high pressure revealed by ultrafast optical spectroscopy, Nature Com- munications 15, 10408 (2024)

    work page 2024

  2. [2]

    Z. Dong, M. Huo, J. Li, J. Li, P. Li, H. Sun, L. Gu, Y. Lu, M. Wang, Y. Wang, and Z. Chen, Visualization of oxy- gen vacancies and self-doped ligand holes inLa3Ni2O7−δ, Nature 630, 847 (2024)

    work page 2024

  3. [3]

    B. Chen, H. Zhang, J. Li, D. Hu, M. Huo, S. Wang, C. Xi, Z. Wang, H. Sun, M. Wang, and B. Shen, Unveil- ing the multiband metallic nature of the normal state in the nickelateLa3Ni2O7, Physical Review B111, 054519 (2025)

    work page 2025

  4. [4]

    T. Xie, M. Huo, X. Ni, F. Shen, X. Huang, H. Sun, H. C. Walker, D. Adroja, D. Yu, B. Shen, L. He, K. Cao, and M. Wang, Strong interlayer magnetic exchange coupling in La3Ni2O7−δ revealed by inelastic neutron scattering, Science Bulletin 69, 3221 (2024)

    work page 2024

  5. [5]

    X. Ren, R. Sutarto, X. Wu, J. Zhang, H. Huang, T. Xi- ang, J. Hu, R. Comin, X. Zhou, and Z. Zhu, Resolving the electronic ground state of La3Ni2O7−δ films, Com- munications Physics 8, 1 (2025)

    work page 2025

  6. [6]

    D. Zhao, Y. Zhou, M. Huo, Y. Wang, L. Nie, Y. Yang, J. Ying, M. Wang, T. Wu, and X. Chen, Pressure-enhanced spin-density-wave transition in double-layer nickelate La3Ni2O7−δ, Science Bulletin 10.1016/j.scib.2025.02.019 (2025)

    work page doi:10.1016/j.scib.2025.02.019 2025

  7. [7]

    J. Yang, H. Sun, X. Hu, Y. Xie, T. Miao, H. Luo, H. Chen, B. Liang, W. Zhu, G. Qu, C.-Q. Chen, M. Huo, Y. Huang, S. Zhang, F. Zhang, F. Yang, Z. Wang, Q. Peng, H. Mao, G. Liu, Z. Xu, T. Qian, D.-X. Yao, M. Wang, L. Zhao, and X. J. Zhou, Orbital-dependent electron correlation in double-layer nickelateLa3Ni2O7, Nature Communications 15, 4373 (2024)

    work page 2024

  8. [8]

    Wang, H.-H

    M. Wang, H.-H. Wen, T. Wu, D.-X. Yao, and T. Xiang, Normal and Superconducting Properties of La3Ni2O7, Chinese Physics Letters41, 077402 (2024)

    work page 2024

  9. [9]

    Z. Liu, H. Sun, M. Huo, X. Ma, Y. Ji, E. Yi, L. Li, H. Liu, J. Yu, Z. Zhang, Z. Chen, F. Liang, H. Dong, H. Guo, D. Zhong, B. Shen, S. Li, and M. Wang, Evi- dence for charge and spin density waves in single crystals of La3Ni2O7 and La3Ni2O6, Science China Physics, Me- chanics & Astronomy66, 217411 (2022)

    work page 2022

  10. [10]

    K. Chen, X. Liu, J. Jiao, M. Zou, C. Jiang, X. Li, Y. Luo, Q. Wu, N. Zhang, Y. Guo, and L. Shu, Evidence of spin density waves inLa3Ni2O7−δ, Phys. Rev. Lett.132, 256503 (2024)

    work page 2024

  11. [11]

    Abadi, K.-J

    S. Abadi, K.-J. Xu, E. G. Lomeli, P. Puphal, M. Isobe, Y. Zhong, A. V. Fedorov, S.-K. Mo, M. Hashimoto, D.-H. Lu, B. Moritz, B. Keimer, T. P. Devereaux, M. Hepting, and Z.-X. Shen, Electronic structure of the alternating monolayer-trilayer phase ofLa3Ni2O7, Phys. Rev. Lett. 134, 126001 (2025)

    work page 2025

  12. [12]

    H. Sun, M. Huo, X. Hu, J. Li, Z. Liu, Y. Han, L. Tang, Z. Mao, P. Yang, B. Wang, J. Cheng, D.-X. Yao, G.-M. Zhang, and M. Wang, Signatures of superconductivity near 80 K in a nickelate under high pressure, Nature621, 493 (2023)

    work page 2023

  13. [13]

    Sakakibara, M

    H. Sakakibara, M. Ochi, H. Nagata, Y. Ueki, H. Saku- rai, R. Matsumoto, K. Terashima, K. Hirose, H. Ohta, M. Kato, Y. Takano, and K. Kuroki, Theoretical anal- ysis on the possibility of superconductivity in the tri- layer ruddlesden-popper nickelateLa4Ni3O10 under pres- sure and its experimental examination: Comparison with La3Ni2O7, Phys. Rev. B109, 1445...

    work page 2024

  14. [14]

    Zhang, C

    M. Zhang, C. Pei, Q. Wang, Y. Zhao, C. Li, W. Cao, S. Zhu, J. Wu, and Y. Qi, Effects of pressure and doping on Ruddlesden-Popper phases Lan+1NinO3n+1, Journal of Materials Science & Technology185, 147 (2024)

    work page 2024

  15. [15]

    G. Wang, N. N. Wang, X. L. Shen, J. Hou, L. Ma, L. F. Shi, Z. A. Ren, Y. D. Gu, H. M. Ma, P. T. Yang, Z. Y. Liu, H. Z. Guo, J. P. Sun, G. M. Zhang, S. Calder, J.-Q. Yan, B. S. Wang, Y. Uwatoko, and J.-G. Cheng, Pressure-induced superconductivity in polycrys- talline La3Ni2O7−δ, Phys. Rev. X14, 011040 (2024)

    work page 2024

  16. [16]

    N. Wang, G. Wang, X. Shen, J. Hou, J. Luo, X. Ma, H. Yang, L. Shi, J. Dou, J. Feng, J. Yang, Y. Shi, Z. Ren, H. Ma, P. Yang, Z. Liu, Y. Liu, H. Zhang, X. Dong, Y. Wang, K. Jiang, J. Hu, S. Nagasaki, K. Kitagawa, S.Calder, J.Yan, J.Sun, B.Wang, R.Zhou, Y.Uwatoko, and J. Cheng, Bulk high-temperature superconductivity in pressurized tetragonal La2PrNi2O7, Na...

    work page 2024

  17. [17]

    J. Li, D. Peng, P. Ma, H. Zhang, Z. Xing, X. Huang, C. Huang, M. Huo, D. Hu, Z. Dong, X. Chen, T. Xie, H. Dong, H. Sun, Q. Zeng, H.-k. Mao, and M. Wang, Identification of superconductivity in bilayer nickelate La3Ni2O7 under high pressure up to 100 GPa, National Science Review , nwaf220 (2025)

    work page 2025

  18. [18]

    Y. Zhou, J. Guo, S. Cai, H. Sun, C. Li, J. Zhao, P. Wang, J. Han, X. Chen, Y. Chen, Q. Wu, Y. Ding, T. Xiang, H.-k. Mao, and L. Sun, Investigations of key issues on 7 the reproducibility of high-Tc superconductivity emerg- ing from compressedLa3Ni2O7, Matter and Radiation at Extremes 10, 027801 (2025)

    work page 2025

  19. [19]

    Hou, P.-T

    J. Hou, P.-T. Yang, Z.-Y. Liu, J.-Y. Li, P.-F. Shan, L. Ma, G. Wang, N.-N. Wang, H.-Z. Guo, J.-P. Sun, Y. Uwatoko, M. Wang, G.-M. Zhang, B.-S. Wang, and J.-G. Cheng, Emergence of High-Temperature Supercon- ductingPhaseinPressurized La3Ni2O7 Crystals,Chinese Physics Letters 40, 117302 (2023)

    work page 2023

  20. [20]

    Zhang, D

    Y. Zhang, D. Su, Y. Huang, Z. Shan, H. Sun, M. Huo, K. Ye, J. Zhang, Z. Yang, Y. Xu, Y. Su, R. Li, M. Smidman, M. Wang, L. Jiao, and H. Yuan, High- temperature superconductivity with zero resistance and strange-metal behaviour inLa3Ni2O7−δ, Nature Physics 20, 1269 (2024)

    work page 2024

  21. [21]

    C.-Q. Chen, Z. Luo, M. Wang, W. Wú, and D.-X. Yao, Trilayer multiorbital models ofLa4Ni3O10, Phys. Rev. B 110, 014503 (2024)

    work page 2024

  22. [22]

    L. Wang, Y. Li, S.-Y. Xie, F. Liu, H. Sun, C. Huang, Y. Gao, T. Nakagawa, B. Fu, B. Dong, Z. Cao, R. Yu, S. I. Kawaguchi, H. Kadobayashi, M. Wang, C. Jin, H.- k. Mao, and H. Liu, Structure Responsible for the Su- perconducting State in La3Ni2O7 at High-Pressure and Low-Temperature Conditions, Journal of the American Chemical Society 146, 7506 (2024)

    work page 2024

  23. [23]

    C. D. Ling, D. N. Argyriou, G. Wu, and J. Neumeier, Neutron diffraction study ofLa3Ni2O7: Structural rela- tionships among n = 1, 2, and 3 phases Lan+1NinO3n+1, Journal of Solid State Chemistry152, 517 (2000)

    work page 2000

  24. [24]

    J. Yuan, Q. Chen, K. Jiang, Z. Feng, Z. Lin, H. Yu, G. He, J. Zhang, X. Jiang, X. Zhang, Y. Shi, Y. Zhang, M. Qin, Z. G. Cheng, N. Tamura, Y.-f. Yang, T. Xiang, J. Hu, I. Takeuchi, K. Jin, and Z. Zhao, Scaling of the strange-metal scattering in unconventional superconduc- tors, Nature 602, 431 (2022)

    work page 2022

  25. [25]

    Q.-G. Yang, D. Wang, and Q.-H. Wang, Possibles±-wave superconductivity in La3Ni2O7, Physical Review B108, L140505 (2023)

    work page 2023

  26. [26]

    Y. Shen, M. Qin, and G.-M. Zhang, Effective Bi-Layer Model Hamiltonian and Density-Matrix Renormaliza- tion Group Study for the High-Tc Superconductivity in La3Ni2O7 under High Pressure, Chinese Physics Letters 40, 127401 (2023)

    work page 2023

  27. [27]

    Yang, G.-M

    Y.-f. Yang, G.-M. Zhang, and F.-C. Zhang, Interlayer valence bonds and two-component theory for high-Tc su- perconductivity of La3Ni2O7 under pressure, Phys. Rev. B 108, L201108 (2023)

    work page 2023

  28. [28]

    Liu, J.-W

    Y.-B. Liu, J.-W. Mei, F. Ye, W.-Q. Chen, and F. Yang, s±-wave pairing and the destructive role of apical-oxygen deficiencies inLa3Ni2O7 under pressure, Phys. Rev. Lett. 131, 236002 (2023)

    work page 2023

  29. [29]

    Z.Luo, B.Lv, M.Wang, W.Wú,andD.-X.Yao,High- TC superconductivity inLa3Ni2O7 based on the bilayer two- orbital t-J model, npj Quantum Materials9, 1 (2024)

    work page 2024

  30. [30]

    Y. Gu, C. Le, Z. Yang, X. Wu, and J. Hu, Effective model and pairing tendency in the bilayer Ni-based supercon- ductor La3Ni2O7, Phys. Rev. B111, 174506 (2025)

    work page 2025

  31. [31]

    C. Xia, H. Liu, S. Zhou, and H. Chen, Sensitive depen- dence of pairing symmetry on Ni-eg crystal field splitting in the nickelate superconductorLa3Ni2O7, Nature Com- munications 16, 1054 (2025)

    work page 2025

  32. [32]

    Fan, J.-F

    Z. Fan, J.-F. Zhang, B. Zhan, D. Lv, X.-Y. Jiang, B. Nor- mand, and T. Xiang, Superconductivity in nickelate and cuprate superconductors with strong bilayer coupling, Phys. Rev. B110, 024514 (2024)

    work page 2024

  33. [33]

    Jiang, Z

    K. Jiang, Z. Wang, and F.-C. Zhang, High-Temperature Superconductivity inLa3Ni2O7, Chinese Physics Letters 41, 017402 (2024)

    work page 2024

  34. [34]

    Y. Wang, K. Jiang, Z. Wang, F.-C. Zhang, and J. Hu, Electronic and magnetic structures of bilayer La3Ni2O7 at ambient pressure, Phys. Rev. B110, 205122 (2024)

    work page 2024

  35. [35]

    Y.-H. Tian, Y. Chen, J.-M. Wang, R.-Q. He, and Z.-Y. Lu, Correlation effects and concomitant two-orbitals±- wavesuperconductivityin La3Ni2O7 underhighpressure, Phys. Rev. B109, 165154 (2024)

    work page 2024

  36. [36]

    Zhang, L.-F

    Y. Zhang, L.-F. Lin, A. Moreo, T. A. Maier, and E. Dagotto, Trends in electronic structures ands±-wave pairing for the rare-earth series in bilayer nickelate super- conductor R3Ni2O7, Phys. Rev. B108, 165141 (2023)

    work page 2023

  37. [37]

    Huang, Z

    J. Huang, Z. D. Wang, and T. Zhou, Impurity and vor- tex states in the bilayer high-temperature superconduc- tor La3Ni2O7, Phys. Rev. B108, 174501 (2023)

    work page 2023

  38. [38]

    Zheng and W

    Y.-Y. Zheng and W. Wú,s±-wave superconductivity in thebilayertwo-orbitalhubbardmodel,Phys.Rev.B 111, 035108 (2025)

    work page 2025

  39. [39]

    Su, Bilayer t−J−J ⊥ model and magnetically medi- ated pairing in the pressurized nickelateLa3Ni2O7, Phys

    X.-Z.Qu, D.-W.Qu, J.Chen, C.Wu, F.Yang, W.Li,and G. Su, Bilayer t−J−J ⊥ model and magnetically medi- ated pairing in the pressurized nickelateLa3Ni2O7, Phys. Rev. Lett. 132, 036502 (2024)

    work page 2024

  40. [40]

    C. Lu, Z. Pan, F. Yang, and C. Wu, Interlayer- Coupling-Driven High-Temperature Superconductivity in La3Ni2O7 under Pressure, Physical Review Letters 132, 146002 (2024)

    work page 2024

  41. [41]

    Zhang, L.-F

    Y. Zhang, L.-F. Lin, A. Moreo, T. A. Maier, and E. Dagotto, Structural phase transition, s±-wave pair- ing, and magnetic stripe order in bilayered superconduc- tor La3Ni2O7 under pressure, Nature Communications 15, 2470 (2024)

    work page 2024

  42. [42]

    Sakakibara, N

    H. Sakakibara, N. Kitamine, M. Ochi, and K. Kuroki, Possible high TC superconductivity in La3Ni2O7 under high pressure through manifestation of a nearly half-filled bilayer hubbard model, Physical Review Letters 132, 106002 (2024)

    work page 2024

  43. [43]

    Lechermann, J

    F. Lechermann, J. Gondolf, S. Bötzel, and I. M. Eremin, Electroniccorrelationsandsuperconductinginstabilityin La3Ni2O7 under high pressure, Physical Review B108, L201121 (2023)

    work page 2023

  44. [44]

    R.Jiang, J.Hou, Z.Fan, Z.-J.Lang,andW.Ku,Pressure driven fractionalization of ionic spins results in cuprate- like high-Tc superconductivity in La3Ni2O7, Phys. Rev. Lett. 132, 126503 (2024)

    work page 2024

  45. [45]

    S. Ryee, N. Witt, and T. O. Wehling, Quenched pair breaking by interlayer correlations as a key to super- conductivity inLa3Ni2O7, Phys. Rev. Lett.133, 096002 (2024)

    work page 2024

  46. [46]

    Heier, K

    G. Heier, K. Park, and S. Y. Savrasov, Competing dxy and s± pairing symmetries in superconducting La3Ni2O7: LDA+FLEX calculations, Phys. Rev. B109, 104508 (2024)

    work page 2024

  47. [47]

    Z. Liao, L. Chen, G. Duan, Y. Wang, C. Liu, R. Yu, and Q. Si, Electron correlations and superconductivity in La3Ni2O7 under pressure tuning, Phys. Rev. B108, 214522 (2023)

    work page 2023

  48. [48]

    Varjas, T

    D. Varjas, T. O. Rosdahl, and A. R. Akhmerov, Qsymm: algorithmic symmetry finding and symmetric Hamilto- nian generation, New Journal of Physics 20, 093026 (2018)

    work page 2018

  49. [49]

    M. G. Vergniory, L. Elcoro, C. Felser, N. Regnault, B. A. 8 Bernevig, and Z. Wang, A complete catalogue of high- quality topological materials, Nature566, 480 (2019)

    work page 2019

  50. [50]

    Cano and B

    J. Cano and B. Bradlyn, Band Representations and Topological Quantum Chemistry, Annual Review of Con- densed Matter Physics12, 225 (2021)

    work page 2021

  51. [51]

    F. Tang, H. C. Po, A. Vishwanath, and X. Wan, Compre- hensive search for topological materials using symmetry indicators, Nature 566, 486 (2019)

    work page 2019

  52. [52]

    F. Tang, H. C. Po, A. Vishwanath, and X. Wan, Efficient topological materials discovery using symmetry indica- tors, Nature Physics15, 470 (2019)

    work page 2019

  53. [53]

    Bradlyn, L

    B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature547, 298 (2017)

    work page 2017

  54. [54]

    F. Tang, H. C. Po, A. Vishwanath, and X. Wan, Towards ideal topological materials: Comprehensive database searches using symmetry indicators, Nature 566, 486 (2019)

    work page 2019

  55. [55]

    Elcoro, B

    L. Elcoro, B. J. Wieder, Z. Song, Y. Xu, B. Bradlyn, andB.A.Bernevig,Magnetictopologicalquantumchem- istry, Nature Communications12, 5965 (2021)

    work page 2021

  56. [56]

    Kruthoff, J

    J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and R.- J. Slager, Topological Classification of Crystalline Insu- lators through Band Structure Combinatorics, Physical Review X 7, 041069 (2017)

    work page 2017

  57. [57]

    Slager, A

    R.-J. Slager, A. Mesaros, V. Juričić, and J. Zaanen, The space group classification of topological band-insulators, Nature Physics 9, 98 (2013), publisher: Nature Publish- ing Group

    work page 2013

  58. [58]

    Bouhon, G

    A. Bouhon, G. F. Lange, and R.-J. Slager, Topological correspondence between magnetic space group represen- tations and subdimensions, Phys. Rev. B 103, 245127 (2021)

    work page 2021

  59. [59]

    The Supplemental Material also contains Refs

    See Supplemental Material at [URL will be inserted by publisher] The stardard BdG Hamiltonian used in the main text; the details of the path integral formulation; the details of the original and refined SI groups and the formulas for superconductors; the comprehensive table the symmetry-allowed pairings and the details of the nu- merical results. The Supp...

    work page

  60. [60]

    Z. Luo, X. Hu, M. Wang, W. Wú, and D.-X. Yao, Bilayer two-orbital model ofLa3Ni2O7 under pressure, Physical Review Letters 131, 126001 (2023)

    work page 2023

  61. [61]

    Coleman, Superconductivity andBCS theory, in In- troduction to Many-Body Physics (Cambridge University Press, 2015) p

    P. Coleman, Superconductivity andBCS theory, in In- troduction to Many-Body Physics (Cambridge University Press, 2015) p. 486–541

    work page 2015

  62. [62]

    O. Can, T. Tummuru, R. P. Day, I. Elfimov, A. Damas- celli, and M. Franz, High-temperature topological super- conductivity in twisted double-layer copper oxides, Na- ture Physics 17, 519 (2021)

    work page 2021

  63. [63]

    S. Ono, H. C. Po, and H. Watanabe, Refined symme- try indicators for topological superconductors in all space groups, Science Advances6, eaaz8367 (2020)

    work page 2020

  64. [64]

    S. Ono, Y. Yanase, and H. Watanabe, Symmetry indi- cators for topological superconductors, Physical Review Research 1, 013012 (2019)

    work page 2019

  65. [65]

    S. Ono, Y. Yanase, and H. Watanabe, Symmetry indica- tors for topological superconductors, Phys. Rev. Res.1, 013012 (2019)

    work page 2019

  66. [66]

    Skurativska, T

    A. Skurativska, T. Neupert, and M. H. Fischer, Atomic limit and inversion-symmetry indicators for topological superconductors, Phys. Rev. Res.2, 013064 (2020)

    work page 2020

  67. [67]

    Bzdušek and M

    T. Bzdušek and M. Sigrist, Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems, Physical Review B96, 155105 (2017)

    work page 2017

  68. [68]

    Feng, Nodal higher-order topological superconduc- tivity fromC6-symmetric dirac semimetals, Phys

    G.-H. Feng, Nodal higher-order topological superconduc- tivity fromC6-symmetric dirac semimetals, Phys. Rev. B 110, 174513 (2024)

    work page 2024

  69. [69]

    Coh and D

    S. Coh and D. Vanderbilt, Python tight binding (pythtb) (2022)

    work page 2022

  70. [70]

    H.-X. Xu, Y. Xie, D. Guterding, and Z. Wang, Competi- tion of superconducting pairing symmetries inLa3Ni2O7 (2025)

    work page 2025

  71. [71]

    S. Ono, K. Shiozaki, and H. Watanabe, Classification of time-reversal symmetric topological superconducting phases for conventional pairing symmetries, Phys. Rev. B 109, 214502 (2024)

    work page 2024

  72. [72]

    S. Ono, H. C. Po, and K. Shiozaki,Z2-enriched symme- try indicators for topological superconductors in the 1651 magnetic space groups, Phys. Rev. Res.3, 023086 (2021)

    work page 2021

  73. [73]

    S. Fan, M. Ou, M. Scholten, Q. Li, Z. Shang, Y. Wang, J. Xu, H. Yang, I. M. Eremin, and H.-H. Wen, Su- perconducting gaps revealed by STM measurements on La2PrNi2O7 thin films at ambient pressure (2025), arXiv:2506.01788 [cond-mat.supr-con]

    work page arXiv 2025