Theoretical investigation of the superconducting pairing symmetry in a bilayer two-orbital model of pressurized La$_3$Ni$_2$O$_7$

Yi Gao

REVIEW 3 major objections 4 minor 1 cited by

Theoretical investigation of the superconducting pairing symmetry in a bilayer two-orbital model of pressurized La$_3$Ni$_2$O$_7$

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read In the bilayer two-orbital model of La3Ni2O7, pairing symmetry is decided by magnetic competition: γ band alone gives odd-frequency triplet, four bands give s±-wave, and residual ferromagnetic fluctuation gives d-wave.

desk verdict A useful band-by-band RPA decomposition that explains the s±/d-wave competition in La3Ni2O7, but the headline s± prediction is parameter-set specific and the paper is honest about that. read the letter →

arxiv 2412.11429 v1 pith:IXGM4ESD submitted 2024-12-16 cond-mat.supr-con

Yi Gao This is my paper · ORCID

classification cond-mat.supr-con

keywords La3Ni2O7bilayernickelatepairingsymmetryspinfluctuationsodd-frequencys±-waveEliashbergequationrandomphaseapproximation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to settle why published calculations disagree about the superconducting pairing symmetry in pressurized La$_3$Ni$_2$O$_7$. Working with the bilayer two-orbital model, it adds bands one at a time and shows that the $\gamma$ band is special: it has a small Fermi velocity, generates a strong ferromagnetic spin fluctuation near $q = (0,0)$, and by itself would give an odd-frequency, $s$-wave spin-triplet state. Adding the $\alpha$, $\beta$, and $\delta$ bands suppresses this ferromagnetic response and enhances antiferromagnetic fluctuations, so the full four-band model ends up in a spin-singlet $s_\pm$-wave state, with a sign change between the $\beta$ Fermi surface and the $\alpha$/$\gamma$ surfaces. If the ferromagnetic fluctuation is not fully suppressed, the same calculation instead produces $d$-wave pairing, which the paper identifies as the source of the conflicting results in the literature. The practical point is that the ground-state symmetry is not a fixed property of the material but depends on which magnetic fluctuations dominate.

What carries the argument

The machinery is a band-resolved random-phase-approximation susceptibility feeding a linearized Eliashberg equation. The spin susceptibility $\chi_s(q) = [I - \chi_0(q) U_s]^{-1}\chi_0(q)$ is evaluated for the four-band model, and its largest eigenvalue $\rho_s(q)$ shows where in momentum space the spin fluctuations live; the Stoner factor $\alpha_s$ measures the proximity to a magnetic instability. The paper then artificially keeps only selected bands in the Green's functions, computes the pairing interaction $V(q) = \tfrac{1}{2}[3U_s\chi_s U_s - U_c\chi_c U_c + U_s + U_c]$ for singlet pairing (and the analogous triplet expression), and solves the linearized Eliashberg equation by the power method. The key diagnostic is the location of the $\rho_s(q)$ peak: a ferromagnetic peak at $q=(0,0)$ promotes odd-frequency triplet $s$-wave pairing, while antiferromagnetic peaks at finite $Q$ promote sign-changing singlet pairing. The narrow $\gamma$ band and its small Fermi velocity are what make the ferromagnetic region so influential.

What would settle it

Measure the magnetic response of pressurized La$_3$Ni$_2$O$_7$ with inelastic neutron scattering: the four-band model predicts the dominant spin fluctuation sits at $Q_1 \approx (\pm 0.84\pi, 0)$ and $(0, \pm 0.84\pi)$, not at $q=(0,0)$; a strong ferromagnetic peak near $q=(0,0)$ would falsify the $s_\pm$-wave claim. Alternatively, a phase-sensitive experiment that finds no sign change between the $\beta$ Fermi surface and the $\alpha$ and $\gamma$ Fermi surfaces would rule out $s_\pm$-wave pairing.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a single band, $\gamma$, carries the ferromagnetic instability that decides the pairing channel. Solving the linearized Eliashberg equation with only the $\gamma$ band gives a spin-triplet pairing function with odd frequency dependence ($\Delta_{\gamma\gamma}(k, ip_n) = -\Delta_{\gamma\gamma}(k, -ip_n)$), even parity in momentum, and $s$-wave symmetry; its magnetic driver is a broad $\rho_s(q)$ peak around $q=(0,0)$. When the other bands are switched on, the antiferromagnetic peaks from $\beta$--$\gamma$, $\beta$--$\beta$, and $\alpha$--$\gamma$ scattering grow, while the $\delta$ band, though unoccupied, removes the ferromagnetic response through inter-band scattering. With all four bands, $\rho_s(q)$ peaks at $Q_1 \approx (\pm 0.84\pi, 0)$ and $(0, \pm 0.84\pi)$, and the leading instability is spin-singlet $s_\pm$-wave: the gap is approximately isotropic and of one sign on the $\alpha$ and $\gamma$ sheets, and of the opposite sign on the $\beta$ sheet, with nodes or gap minima near $k_x = \pm k_y$ on $\beta$. Omitting the $\delta$ band leaves the ferromagnetic region intact and turns the leading solution into $d_{x^2-y^2}$-wave with nodes along $k_x=\pm k_y$. The paper therefore claims that the correct pairing symmetry in this model is not unique: it depends on whether the ferromagnetic fluctuation is completely suppressed, which in turn is controlled by the tight-binding parameters.

Load-bearing premise

The $s_\pm$-wave conclusion rests on the specific set of electron hopping parameters taken from Ref. [2]; the paper itself notes that with other published parameter sets the ferromagnetic fluctuation is not fully suppressed and the pairing becomes $d$-wave, so if those alternative parameter sets are equally valid for pressurized La$_3$Ni$_2$O$_7$, the central claim is not robust.

Editorial extensions

If this is right

If the paper is right, models that neglect the unoccupied $\delta$ band will systematically overestimate the tendency to $d$-wave pairing, because the $\delta$ band is needed to suppress the ferromagnetic fluctuation.
The full four-band model predicts an $s_\pm$-wave gap on the $\beta$ Fermi surface with nodes or near-nodes along $k_x = \pm k_y$, while the $\alpha$ and $\gamma$ sheets carry a relatively isotropic gap of the opposite sign.
A material realization that isolates the $\gamma$ band would be a candidate for odd-frequency $s$-wave spin-triplet superconductivity, but in the full model this state loses to singlet pairing.
Experiments that determine the gap symmetry (such as specific heat, penetration depth, or phase-sensitive probes) should also be able to discriminate between the competing tight-binding parameter sets.
The pairing mechanism here is magnetic in origin: whichever spin fluctuation dominates sets the pairing channel, so magnetic probes and superconducting gap probes should agree.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the competition story is right, artificially flattening the $\gamma$ band (by reducing its hopping) should strengthen the ferromagnetic response and push the leading pairing toward the triplet/$d$-wave side; this is a testable consequence the paper does not state.
The same band-resolved mechanism may transfer to other bilayer nickelates with the $R_3$Ni$_2$O$_7$ structure, where modest changes in band structure could flip the pairing symmetry between $s_\pm$-wave and $d$-wave.
A heterostructure or exfoliated flake that selects only the $\gamma$-like band could be a route to realizing odd-frequency spin-triplet pairing, though the paper proposes no such device.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth a look. The new thing here is the case-by-case band decomposition: take the bilayer two-orbital model, keep only selected bands in the Green's function, and see what pairing symmetry each combination produces. That is a genuinely useful diagnostic that I don't think anyone did systematically for this material. The main result is that the γ band alone produces ferromagnetic spin fluctuations and an odd-frequency s-wave triplet, and adding the other bands suppresses those FM fluctuations, eventually giving s±. The full-model s± itself is not new (Refs. 3 and 9 already have it), and the paper says so. What's new is the mechanism picture and the observation that the unoccupied δ band matters.

The calculation looks competently done. Standard RPA plus linearized Eliashberg, 64×64 grid, Matsubara sums, and they give explicit fits for the pairing functions in each case. No code or data, but the methods are specified well enough to re-implement. The paper is also honest at the end: it admits that with the tight-binding parameters of Refs. 21 and 22 you get d-wave instead of s±. That is the main soft spot. The abstract says 'finally into an s±-wave one' with more confidence than the evidence supports, because the result is tied to one of several published parameter sets and the paper offers no criterion for why that set is the right one. The mechanism—competition between FM fluctuations from the narrow γ band and AFM from the others—may well be robust, but the sign of the prediction is not.

Two smaller concerns. First, each case uses a different U, chosen to bring the Stoner factor to the same value near 0.9. That makes the 'evolution' across cases cleaner, but it means you are comparing calculations at different interaction strengths. The paper does say the qualitative conclusions hold at U=1.16, but the eigenvalues drop to ~0.1, so the triplet preference is weak at the physical U. Second, the band projection is done by zeroing out unitary-matrix elements in the Green's function, which is a numerical diagnostic rather than a physical process. That's fine for what it is, but it shouldn't be over-read as 'this is what that band alone would do in the real material.'

The citation pattern is clean; the self-citation to Ref. 25 is for standard RPA formulas. No circularity. I'd send this to peer review. The right referee request is to tighten the abstract to match the parameter-set dependence and to show at least one fixed-U comparison across the cases, even at the cost of smaller eigenvalues. The paper deserves a serious referee; it's a useful contribution to a crowded literature.

Referee Report

3 major / 4 minor

Summary. This paper investigates the superconducting pairing symmetry of pressurized La3Ni2O7 using a bilayer two-orbital tight-binding model and RPA/linearized-Eliashberg calculations. The model has four bands (α, β, γ, δ), and the paper analyzes the pairing obtained when different subsets of these bands are retained in the Green's functions. The central result is that the γ band alone produces ferromagnetic spin fluctuations and an odd-frequency, s-wave spin-triplet pairing state, while adding the other bands suppresses the ferromagnetic fluctuations and enhances antiferromagnetic ones, leading to a d-wave singlet and finally to an s±-wave singlet when all four bands are included. The paper attributes the conflicting s±-versus-d-wave predictions in the literature to whether this ferromagnetic fluctuation is completely suppressed, which depends on the tight-binding parameters.

Significance. If the mechanism is correct, the paper offers a physically transparent explanation for the spread of pairing-symmetry predictions in pressurized La3Ni2O7: the competition between ferromagnetic fluctuations from the narrow γ band and antiferromagnetic fluctuations from other bands. The methodological backbone is standard RPA spin-fluctuation theory and linearized Eliashberg equations, implemented with a dense momentum/frequency grid, and the full-model result (Case 1) reproduces earlier published s±-wave solutions for the same tight-binding parameters. The paper is also commendably explicit in the Summary that the s±-wave outcome is tied to the specific parameter set of Ref. 2, and that alternative published parameter sets (Refs. 21, 22) lead to d-wave pairing. The main weakness is that this parameter-set sensitivity is not reflected in the abstract's unqualified 'finally into an s±-wave one,' and the cross-case comparisons rely on an ad-hoc normalization of the interaction strength.

major comments (3)

[Abstract; Section IV (Summary)] The abstract states that the pairing 'finally into an s±-wave one' as the outcome for pressurized La3Ni2O7, but the Summary explicitly concedes that with the tight-binding parameters of Refs. 21 and 22 the same analysis yields d-wave pairing because the ferromagnetic fluctuation is not completely suppressed. Since the paper offers no physical criterion for selecting the Ref. 2 parameter set over the alternatives, the headline prediction is parameter-set specific, not a robust statement about the material. The abstract and title should be qualified (e.g., 'for the parameter set of Ref. 2') or the paper should justify why that parameter set is the appropriate representation of pressurized La3Ni2O7.
[Section III, Cases 2-6] The comparison of pairing tendencies across the band-selection cases is made by tuning U to different values (U=4, 2.08, 1.7, 1.5, 2) so that the Stoner factor α_s≈0.9 in each case, even though the physical interaction in the full model is U=1.16. This normalization is an assumption rather than a derived result, and the relative strength of ferromagnetic versus antiferromagnetic fluctuations could differ at fixed U. Although the paper states that the conclusions of Cases 3-6 are qualitatively unchanged at U=1.16, the central mechanistic claim would be more convincing if the paper showed the evolution of the leading pairing symmetry as a function of U within each case, or justified why α_s≈0.9 is the correct common comparison point.
[Section II, Eq. (6); Section III, Cases 2-6] The procedure of 'artificially setting Q_{s1 s2}=0' to isolate individual bands in the Green's function is not a standard band projection: it discards selected columns of the unitary transformation rather than truncating the Hilbert space, and it is not demonstrated that the resulting Green's function corresponds to any physical Hamiltonian or preserves the analytic behavior required for the Eliashberg equation. Since the case-by-case results are the main evidence for the proposed mechanism, this projection should be justified more rigorously or shown to be equivalent to a well-defined restricted model (e.g., an orbital-selective or band-selective decoupling).

minor comments (4)

[Section III, Case 6, Eq. (24)] In Eq. (24), the term 'cos 4x cos 3ky' should presumably be 'cos 4kx cos 3ky'.
[References, Ref. 29] Reference 29 contains a corrupted author name 'A. T. R ϕer'; this should be corrected to 'A. T. Rømer'.
[Section III, Case 2] The pairing on the β band is described as 's-wave' even though the text states there are nodes along kx=±ky and at kx=±π, ky=±π; the classification would be clearer if the point-group symmetry were stated explicitly, since a nodal s-wave is an unusual characterization.
[Section I, Introduction] The phrase 'In literatures, they are denoted as' should be 'In the literature, they are denoted as'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the s±/d-wave pairing symmetry is an output of RPA susceptibility and Eliashberg eigenvalue analysis, not an input or fit.

full rationale

The derivation chain is self-contained and the central claim is computed, not imposed. The model Hamiltonian (Eq. 1) and its tight-binding parameters are taken from Ref. 2, and the interaction (Eq. 7) is the standard multiorbital Hubbard form. The bare susceptibility (Eq. 8), RPA susceptibilities (Eq. 9), and the linearized Eliashberg equation (Eq. 12) are standard formulas; Ref. 25 is one citation among several (Refs. 23, 24, 26, 28) for these equations, so the self-citation is not load-bearing. The pairing symmetry in each case is obtained by numerically solving Eq. (12) for the largest positive eigenvalue and reading off the anomalous self-energy (Eq. 17); no target symmetry is inserted into the calculation. The case-by-case band selection is a controlled decomposition achieved by zeroing unitary-matrix elements in Eq. (6), which is a numerical probe of band contributions rather than an assumption of the outcome. Tuning U in the reduced-band cases to reach a comparable Stoner factor α_s ≈ 0.9 changes the interaction strength but not the symmetry; the paper explicitly states that at U = 1.16 the conclusions do not change qualitatively. The trigonometric fits in Eqs. (18)-(24) are post hoc descriptions of the converged gap, not inputs. The Summary's statement that different published tight-binding parameters (Refs. 21, 22) lead to d-wave pairing is an admission of parameter sensitivity, not a circular reduction. No fitted parameter is renamed as a prediction, and no uniqueness or ansatz is imported from the author's prior work. Therefore there is no significant circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The calculation rests on standard RPA spin-fluctuation theory, a specific tight-binding parameterization, and a band-projection method introduced for this analysis. The only fitted quantity is the interaction U (and the derived target α_s), which is tuned per case. No new physical entities are postulated.

free parameters (2)

U (intra-orbital Coulomb repulsion) = 1.16 eV (case 1), 4 eV (case 2), 2.08 eV (case 3), 1.7 eV (case 4), 1.5 eV (case 5), 2 eV (case 6)
U is chosen in each case to reach the target Stoner factor α_s≈0.9, except case 1 which uses the literature value from Ref. 3. The pairing symmetry and its evolution across cases depend on these choices.
Target Stoner factor α_s = 0.9
The paper tunes U in Cases 2-6 so that α_s≈0.9, matching the proximity to magnetic instability in Case 1. This hand-chosen normalization affects the comparison of pairing eigenvalues across cases.

assumptions (5)

domain assumption The RPA susceptibilities (Eq. 9) and the Eliashberg pairing interaction (Eqs. 13-14) correctly describe the leading superconducting instability in this multiorbital system.
The paper's entire calculation is built on standard RPA spin-fluctuation theory (Refs. 23-26). This is the accepted weak-coupling framework for this material class.
domain assumption The bilayer two-orbital tight-binding model (Eq. 1) with parameters from Ref. 2 accurately represents pressurized La3Ni2O7.
The central s±-wave conclusion is contingent on these parameters. The paper itself notes in the Summary that other published parameter sets (Refs. 21, 22) yield d-wave pairing.
domain assumption Interaction parameters satisfy U' = U - 2J_H and J_H = J' = U/6.
These relations are assumed in the multiorbital Hubbard interaction (Eq. 7) and are common but not universal choices; they fix the relative strength of different interaction channels.
ad hoc to paper Setting Q matrix elements to zero isolates the contribution of a chosen band (or bands) to the Green's function and pairing.
Used to define Cases 2-6 in Section III. The resulting approximate Green's functions do not correspond to any physical truncation of the model, so the 'band-only' results are interpretative rather than directly measurable.
ad hoc to paper Comparing cases at a fixed Stoner factor α_s≈0.9 by adjusting U is a valid normalization for assessing pairing tendencies.
The paper tunes U per case to reach the same proximity to magnetic order. This changes the interaction strength and may bias the relative pairing eigenvalues; it is a methodological choice, not a physical constraint.

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Pith. "Pith review of Theoretical investigation of the superconducting pairing symmetry in a bilayer two-orbital model of pressurized La$_3$Ni$_2$O$_7$." pith.science (2026). https://pith.science/paper/IXGM4ESD

@misc{pith2026241211429,
  author       = {Pith},
  title        = {Pith review of: Theoretical investigation of the superconducting pairing symmetry in a bilayer two-orbital model of pressurized La$_3$Ni$_2$O$_7$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IXGM4ESD}},
  note         = {Machine review of arXiv:2412.11429}
}

abstract

We investigate the superconducting pairing symmetry in pressurized La$_3$Ni$_2$O$_7$ based on a bilayer two-orbital model. There are two symmetric bands $\alpha$ and $\gamma$, as well as two antisymmetric ones $\beta$ and $\delta$. It is found that the $\gamma$ band induces considerable ferromagnetic spin fluctuation and prefers an odd-frequency, $s$-wave spin triplet pairing state. The addition of the other bands gradually suppresses the ferromagnetic spin fluctuation and enhances the antiferromagnetic ones. The superconducting pairing then evolves from a spin triplet into a $d$-wave spin singlet, and finally into an $s_\pm$-wave one. The competition between the $d$-wave and $s_\pm$-wave pairings relies on whether the ferromagnetic spin fluctuation is suppressed completely or not.

Figures

Figures reproduced from arXiv: 2412.11429 by the authors.

**Figure 2.** FIG. 2: (a) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3: (a) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4: (a) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5: (a) [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6: (a) [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7: (a) [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Probing Sign-Changing Order Parameters via Impurity States in unconventional superconductors: Implications for La$_3$Ni$_2$O$_7$ Superconductors with interlayer pairing
cond-mat.supr-con 2025-01 conditional novelty 5.0 of 10

Impurity-induced in-gap states in superconductors are tied to sign-changing order parameters, and the effect can be used to test pairing symmetry in La3Ni2O7.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

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 80K in a nickelate under high pressure, Nature 621, 493-498 (2023)

work page 2023

[2] [2]

Z. Luo , X. Hu , M. Wang , W. W u , and D.-X. Yao, Bilayer Two-Orbital Model of La _3 Ni _2 O _7 under Pressure, Phys. Rev. Lett. 131, 126001 (2023)

work page 2023

[3] [3]

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 in La _3 Ni _2 O _7 under Pressure, Phys. Rev. Lett. 131, 236002 (2023)

work page 2023

[4] [4]

Y. Gu, C. Le, Z. Yang, X. Wu, and J. Hu, Effective model and pairing tendency in bilayer Ni-based superconductor La _3 Ni _2 O _7 , arXiv:2306.07275v2

arXiv

[5] [5]

Yang, G.-M

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