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Pairing symmetry and superconductivity in La$_3$Ni$_2$O$_7$ thin films

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A renormalized mean-field calculation on a bilayer two-orbital t-J model predicts s±-wave pairing in La3Ni2O7 thin films, reproducing the measured nodeless gap on the beta Fermi pocket and a transition temperature near 60 K.

desk verdict A credible RMFT transfer to thin-film LNO that likely has the right pairing symmetry; the new inter-orbital d-wave channel is the real novelty, and the μ sensitivity deserves a robustness check before signing off. read the letter →

arxiv 2506.20727 v3 pith:3OSNT5Z7 submitted 2025-06-25 cond-mat.supr-con cond-mat.str-el

Wenyuan Qiu , Zhihui Luo , Xunwu Hu , Dao-Xin Yao This is my paper · ORCID
classification cond-mat.supr-concond-mat.str-el PACS 74.20.Rp74.78.Fk
keywords nickelatesuperconductorsLa3Ni2O7thinfilmss±-wavepairingsymmetryrenormalizedmean-fieldtheorybilayertwo-orbitalt-JmodelinterlayersuperexchangenodelesssuperconductinggapFermi-surfacepockets
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 argues that the ambient-pressure superconductivity discovered in La$_3$Ni$_2$O$_7$ thin films ($T_c$ above 40 K) shares the $s_\pm$-wave pairing symmetry of pressurized bulk La$_3$Ni$_2$O$_7$ — the superconducting state whose gap changes sign between different Fermi pockets. The authors obtain this by solving a bilayer two-orbital $t$–$J$ model with a renormalized mean-field method, and they trace the pairing to the strong interlayer magnetic superexchange between $d_{z^2}$ orbitals of the two Ni layers. Their calculation reproduces, at least roughly, the observed nodeless shape of the superconducting gap on the $\beta$ Fermi pocket and a transition temperature near 60 K, in line with the measured 40–60 K range. They also find an in-plane inter-orbital pairing component with a $d$-wave form factor that coexists with the dominant $s_\pm$ channel and enhances it, a feature specific to the $I4/mmm$ lattice of this nickelate. If the calculation is right, the same superconducting mechanism that operates in pressurized bulk La$_3$Ni$_2$O$_7$ is at work at ambient pressure in the films, so the pairing physics can be studied directly with surface probes.

What carries the argument

The engine is a bilayer two-orbital $t$–$J$ model built from $d_{z^2}$ and $d_{x^2-y^2}$ orbitals on two Ni layers, with tight-binding terms taken from a DFT simulation of the double-stacked thin film and spin-exchange couplings $J_\perp$, $J_\parallel$, $J_{xz}$ plus a Hund's coupling $J_H$. The load-bearing coupling is the interlayer superexchange $J_\perp = 0.135$ eV between $d_{z^2}$ orbitals, estimated by exact diagonalization of a five-site Ni–O chain and about 25% smaller than the bulk value 0.18 eV; it generates the dominant interlayer pairing bond $\Delta^z_\perp$. The model is solved by renormalized mean-field theory (RMFT): the Gutzwiller-projected spin-exchange terms become self-consistently determined pairing bonds $\Delta^{\mu\nu}_{ij}$, and the relative phases of those bonds fix the pairing symmetry. In the bonding-antibonding basis of the two layers, the gap on each pocket is a sum of an interlayer term and an in-plane $\cos k_x + \cos k_y$ term, and the requirement to maximize the gap on the $\gamma$ pocket forces the in-plane $d_{z^2}$ bond to sit antiphase to the interlayer bonds. The supplementary material shows the pairing matrix lives in the $A_{1g}$ irrep, where an off-diagonal inter-orbital bond with the $d$-wave form factor $\cos k_x - \cos k_y$ coexists with the $s_\pm$ diagonal bonds.

What would settle it

Resolve the gap phase on the $\beta$ and $\gamma$ pockets: the $s_\pm$ claim requires both pockets to be nodeless with opposite signs, so a phase-sensitive measurement (corner Josephson junction, quasiparticle interference) or a high-resolution ARPES scan that finds a node on the $\beta$ pocket along the Brillouin-zone diagonal would rule it out. A cheaper computational check is to rerun the RMFT at the undoped DFT filling instead of the tuned $\mu = -0.05$ eV and see whether the $s_\pm$-wave and the nodeless $\beta$-pocket gap persist, since the paper's chemical-potential choice is the load-bearing input.

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Extended reading notes

Core claim

The paper's central claim is that renormalized mean-field theory on a bilayer two-orbital $t$–$J$ model produces an $s_\pm$-wave superconducting state for La$_3$Ni$_2$O$_7$ thin films: the gap on the $\beta$ and $\gamma$ Fermi-surface pockets is nodeless and of opposite sign, while the $\alpha$ pocket carries a sign flip and a node. The dominant pairing bond is the interlayer $d_{z^2}$ spin singlet $\Delta^z_\perp$, generated by the strong interlayer superexchange $J_\perp \approx 0.135$ eV, and the resulting gap shape on the $\beta$ pocket and the mean-field $T_c \approx 60$ K roughly match the ARPES nodeless gap and the experimental $T_c$ of 40–60 K. Analyzing the orbital-resolved pairing bonds projected onto the Fermi surface, the paper attributes the nodeless feature of the $\beta$ pocket to interlayer pairing within both $d_{z^2}$ and $d_{x^2-y^2}$ orbitals acting with the same sign, and it identifies an in-plane inter-orbital pairing bond $\Delta^{xz}_\parallel$ between $d_{z^2}$ and $d_{x^2-y^2}$ whose $d$-wave form factor places it in the same $A_{1g}$ symmetry channel as the dominant $s_\pm$ pairing, so the two channels cooperate instead of competing. The paper states this pairing structure should be described as the $s_\pm$-wave, with a phase arrangement $\mathrm{sgn}\,\Delta^z_\perp = \mathrm{sgn}\,\Delta^x_\perp = \mathrm{sgn}\,\Delta^x_\parallel = -\mathrm{sgn}\,\Delta^z_\parallel$ that follows from maximizing the total gap magnitude on each pocket.

Load-bearing premise

The calculation inherits its band structure, Fermi-surface geometry, and filling from a DFT-based tight-binding model of the double-stacked thin film, with the chemical potential set to $\mu = -0.05$ eV (fillings $n_z = 0.8$, $n_x = 0.58$) to better agree with the experiments; if that input is not representative of the real films, the predicted $s_\pm$ pattern on specific pockets, the nodeless $\beta$ gap, and $T_c \approx 60$ K would not survive.

Editorial extensions

If this is right

  • The ambient-pressure thin films display the same $s_\pm$-wave pairing symmetry as pressurized bulk La$_3$Ni$_2$O$_7$, so the high-pressure phase can be studied at ambient pressure with ARPES, STM, and other surface probes.
  • The drop from about 80 K in the pressurized bulk to about 60 K in the films is attributed to a roughly 25% reduction of the interlayer superexchange $J_\perp$ (0.18 eV to 0.135 eV), consistent with the simple estimate $J_\perp \propto (t_z^\perp)^2$.
  • The nodeless $\beta$-pocket gap measured by ARPES is produced jointly by interlayer pairing in both $d_{z^2}$ and $d_{x^2-y^2}$ orbitals, so experiments that track a single orbital channel would not see the full pairing state.
  • The in-plane inter-orbital $d$-wave pairing $\Delta^{xz}_\parallel$ enhances the dominant interlayer $s_\pm$ channel, and this coexistence is a symmetry-stabilized feature of the La$_3$Ni$_2$O$_7$ lattice rather than a numerical accident.
  • The large gap on the $\gamma$ pocket, about five times the $\alpha$-pocket gap, may explain why the $\gamma$ sheet is visible in normal-state ARPES but absent in superconducting-state measurements.

Reading between the lines

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

  • By the paper's logic that the pairing preference is fixed by the Fermi-surface topology plus $J_\perp$, doping or strain that moves the $\beta$ or $\gamma$ pockets through the Fermi level should flip the relative gap signs or destroy the nodeless state — a testable doping–strain phase map the authors do not draw.
  • The same RMFT treatment could be run for the (La,Pr)$_3$Ni$_2$O$_7$ films, also reported superconducting above 40 K; whether rare-earth substitution preserves the roughly 0.135 eV interlayer exchange would test whether the mechanism is identical across film compositions.
  • Because the authors state that $J_\perp$ may set only an upper bound on $T_c$, a continuous strain scan across substrates with different lattice mismatch could map $T_c$ against the computed $J_\perp(c/a)$ and reveal which additional factors — hybridization, density waves, interlayer Josephson coupling — cap the transition temperature.
  • If the $\gamma$-pocket gap really exceeds the detection window of superconducting-state ARPES, temperature-dependent measurements across $T_c$ should make the sheet re-emerge in the normal state, turning the paper's explanation of the invisible pocket into a direct experimental check.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies superconductivity in La3Ni2O7 thin films using a bilayer two-orbital t-J model with DFT-based tight-binding parameters for the double-stacked film and superexchange couplings (J⊥=0.135 eV, J||=0.084 eV, Jxz=0.03 eV) estimated from exact diagonalization. Solving the model with renormalized mean-field theory at a chemical potential μ=-0.05 eV (fillings nz=0.8, nx=0.58), the authors report an s±-wave pairing state whose gap is nodeless on the β pocket and opposite in sign on the γ pocket, with a mean-field Tc≈60 K, in qualitative agreement with recent ARPES and transport experiments on thin films. The paper further analyzes orbital-resolved pairing bonds and identifies an in-plane interorbital d-wave component that cooperates with the dominant interlayer dz2 pairing.

Significance. If the result is robust, it extends the s± pairing scenario from pressurized bulk La3Ni2O7 to ambient-pressure thin films, providing a concrete microscopic mechanism for the experimentally observed nodeless β-pocket gap and the two-gap STM data. A methodological strength is that the superexchange couplings are derived from DFT-based exact diagonalization rather than fitted to the superconducting Tc, and the s± pattern emerges from the self-consistent RMFT solution. The paper also makes a falsifiable prediction: the γ pocket should carry a large gap roughly five times the α-pocket gap, which can be tested by ARPES and STM. The main weakness is the lack of sensitivity analysis for the chemical potential and tight-binding inputs, which is addressed in the major comments.

major comments (3)
  1. [Model and method; Fig. 1(a); Fig. 2] The entire pocket-resolved result is computed at a single chemical potential μ=-0.05 eV, chosen 'to better agree with the experiments,' with no sensitivity analysis. The Fermi surface topology and the orbital character of the α, β, γ pockets are inputs from the tight-binding model of Ref. [57], and the sign structure, nodelessness, and Tc are all computed on that Fermi surface. If μ were varied over a plausible range (e.g., ±0.1 eV), or if the interlayer hopping tz⊥ (which controls J⊥) were changed, the β pocket could shift relative to the other pockets or change orbital character, potentially altering the sign assignment. The paper does not report a phase diagram vs μ or filling, nor does it overlay the model Fermi surface on the ARPES Fermi surfaces of Refs. [56,60] to validate the input. Because the same experimental data are used both to pin μ and to claim agreement for the β-pocket gap, the agreement is partially circular. Please add a robustness study over μ and over tz⊥ (or J⊥), and a quantitative Fermi-surface comparison.
  2. [Model and method] The derivation of J|| and Jxz is not described. The text details only the 5-site ED for J⊥; for J||=0.084 eV and Jxz=0.03 eV, the cluster geometry, the p-d hoppings, and the definition of the relevant singlet-triplet gaps are not given. Since these couplings control the in-plane pairing bonds (and Jxz generates the interorbital d-wave channel that is central to the enhancement claim), the values need to be documented or referenced to a reproducible calculation. In addition, the statement that JH=1 eV has 'no significant impact' is made without a supporting figure or table.
  3. [Discussion] The assertion that 'the s±-wave is quite robust under various approaches from weak to strong couplings' is made without citation or supporting calculation. This is precisely the robustness claim needed to address the sensitivity of the result to model details. Please either provide the evidence (e.g., a comparison with weak-coupling RPA or functional RG results for the same model) or temper the statement to reflect the evidence actually shown.
minor comments (4)
  1. [Throughout] There are several typos: 'bule' in the Fig. 1 caption, 'thim films' in the introduction, 'claculated' in the Model and method section, and 'Numbu', 'aviod', 'obove', and 'coorporation' in the supplementary material.
  2. [Fig. 2] Panel (a) lacks a color bar and does not specify the units of the plotted quantity gtΔ; please add a color bar and state the energy scale.
  3. [Fig. 3 and supplementary] The symbols Δz⊥, Δx⊥, Δz||, Δx||, and Δxz|| are introduced without a single defining table; please define all pairing-bond notation in one place, and write the renormalization factors as g_t^z, g_t^x for readability.
  4. [Supplementary, Eq. (7)] The value T x⊥,k = 0.005 (a constant) for the interlayer dx2-y2 hopping is surprisingly small and should be justified, since it underlies the smallness of Δx⊥.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor circularity: the chemical potential is tuned to experiments and then the pocket-resolved gap is compared to ARPES, but the s±-wave result itself is an emergent RMFT output.

  1. fitted input called prediction [Model and method, second paragraph; Pairing symmetry and superconducting gap, paragraph on Fig. 2(c)]
    "a small electron doping level with chemical potential µ = −0.05 is used in our calculation to better agree with the experiments, which gives the electron filling nz=0.8, nx=0.58. ... the gap shape as well as its magnitude around the BZ diagonal direction from our result are generally coordinated with the ARPES result [60]."

    The chemical potential is adjusted to 'better agree with the experiments' before the RMFT run, and this choice controls the β pocket's location, size, and orbital content. The later statement that the computed β-pocket gap is 'coordinated with the ARPES result [60]' therefore uses experimental input to fix the Fermi surface on which the gap is computed, making the agreement a partially circular validation. The circularity is only partial because the s± sign structure, the nodelessness condition, and Tc≈60 K are self-consistent RMFT outputs rather than parameters fitted to the gap data, so the central pairing-symmetry claim is not forced by the fit.

full rationale

The derivation chain is: DFT-derived tight-binding Hamiltonian (parameters reproduced in the supplement) → exact-diagonalization estimates of J⊥, J||, Jxz from the same DFT hoppings → self-consistent RMFT solution that determines the pairing bonds and their phases. The s±-wave assignment and the pocket-resolved nodelessness are emergent: they come from minimizing the free energy, not from presetting the order parameters. The only externally tuned knob is μ = −0.05, chosen to match experimental fillings; this is a legitimate but unquantified input, and the absence of a μ-sensitivity analysis is a robustness concern rather than a demonstration that the prediction is forced. The self-citations to Refs. [7], [8], and [57] are not load-bearing in a circular way because the tight-binding parameters are listed in the supplementary material and the ED/RMFT methods are standard, reproducible, and externally checkable. Overall, the central claim has independent content; the score of 2 reflects the one partially circular comparison between a μ-tuned Fermi surface and the ARPES gap used as confirmation.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claims rest on a model Hamiltonian with couplings derived from DFT-based ED, a mean-field approximation, and an experimental comparison. The only overtly fitted number is the chemical potential; the exchange couplings are derived rather than fitted to superconductivity, so they are listed as axioms with domain assumptions. No new physical particles, forces, or dimensions are introduced.

free parameters (2)
  • Chemical potential mu (electron doping) = -0.05 eV (fillings nz=0.8, nx=0.58)
    Chosen in Model and method to 'better agree with the experiments'; sets the Fermi surface and therefore the pocket-resolved gap features and Tc.
  • Hund coupling JH = 1 eV
    Taken as a realistic value from prior literature, not derived in this paper; the authors state it has no significant impact, but it is still a hand-chosen model input.
assumptions (6)
  • domain assumption The bilayer two-orbital t-J model with dz2 and dx2-y2 orbitals captures the essential low-energy physics of La3Ni2O7 thin films.
    All results follow from this model, defined in Eq. (1)-(3), with no derivation from a more complete many-body Hamiltonian.
  • domain assumption The tight-binding Hamiltonian obtained by averaging over double stacks of the DFT one-UC model (Ref [57]) is accurate for thin-film La3Ni2O7.
    Supplementary Eq. (6)-(7) and Fig. 1(a); the Fermi surface and all gap projections inherit these hoppings.
  • domain assumption Superexchange couplings J⊥=0.135 eV, J||=0.084 eV, Jxz=0.03 eV estimated from a 5-site exact-diagonalization chain with DFT hoppings are representative of the thin film.
    Model and method section; these values set the magnitude of pairing bonds and Tc, but the ED geometry is simplified and no uncertainty is given.
  • standard math Renormalized mean-field theory with Gutzwiller factors gt and gJ is a valid approximation for this strongly correlated t-J model.
    The paper follows Ref [7] and cites Gutzwiller projection; this is a well-known but uncontrolled variational mean-field approach.
  • domain assumption Pairing tendencies of bonds not driven by an explicit exchange term, such as Δx_⊥, are meaningful when read from the mean-field wavefunction.
    Model and method section states these values 'reflect the pairing tendency towards a global coherent state'; this interpretation is not independently proven.
  • standard math The A1g channel can host coexisting s±-wave and d-wave form factors as classified in Ref [65].
    Supplementary Eq. (8)-(9) and the discussion invoke group theory from a preprint; the coexistence is asserted as robust by symmetry.

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Cite this review

Pith. "Pith review of Pairing symmetry and superconductivity in La$_3$Ni$_2$O$_7$ thin films." pith.science (2026). https://pith.science/paper/3OSNT5Z7

@misc{pith2026250620727,
  author       = {Pith},
  title        = {Pith review of: Pairing symmetry and superconductivity in La$_3$Ni$_2$O$_7$ thin films},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3OSNT5Z7}},
  note         = {Machine review of arXiv:2506.20727}
}
abstract

The recent discovery of superconductivity with a transition temperature $T_c$ over 40 K in La$_3$Ni$_2$O$_7$ and (La,Pr)$_{3}$Ni$_2$O$_7$ thin films at ambient pressure marks an important step in the field of nickelate superconductors. Here, we perform a renormalized mean-field theory study of the superconductivity in $\mathrm{La_3Ni_2O_7}$ thin films, using a bilayer two-orbital $t-J$ model. Our result reveals an $s_\pm$-wave pairing symmetry driven by the strong interlayer superexchange coupling of $d_{z^2}$ orbital, resembling the pressurized bulk case. Also, we roughly reproduce the experimentally observed nodeless shape of the superconducting gap at the $\beta$ pocket and the superconducting $T_c$. In addition, by analysing the orbital-resolved pairing configurations and their projections onto Fermi surface, we find that the nodeless feature of $\beta$ pocket is related to the interlayer pairing within both $d_{z^2}$ and $d_{x^2-y^2}$ orbitals. Moreover, we identify a formation of the inplane inter-orbital $d$-wave pairing between $d_{z^2}$ and $d_{x^2-y^2}$ orbitals, which can even enhance the dominated interlayer $s_\pm$-wave. Our study particularly highlights the diverse relations of different pairing channels in $\mathrm{La_3Ni_2O_7}$ that holds a complex Fermi surface.

Figures

Figures reproduced from arXiv: 2506.20727 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Band structure of the 4-orbital tight-binding [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. (b-d) show the absolute value gt|∆| as a function of angle θ for α, β and γ pockets, respectively. For α pocket in [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Orbital-resolved superconducting energy gap on FS [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Superconducting gap as a function of temperature for [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

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

  1. Superexchanges and Charge Transfer in the La$_3$Ni$_2$O$_7$ Thin Films

    cond-mat.supr-con 2025-11 conditional novelty 6.0 of 10

    In La3Ni2O7 thin films, the interlayer d3z2-r2 antiferromagnetic coupling is about 27% weaker than in bulk, in-plane coupling is nearly unchanged, and hole/electron doping is particle-hole asymmetric.

  2. Symmetry-Based Microscopic Theory of the Unconventional Pairing Mechanism in La$_5$Ni$_3$O$_{11}$

    cond-mat.supr-con 2026-07 conditional novelty 5.0 of 10

    La5Ni3O11 superconductivity is predicted to be a two-gap s± state in the bilayer subsystem, with the T_c reduction tied to a reduced interlayer-to-intralayer hopping ratio.

  3. Compressive Strain Turns $s^{\pm}$ into $d$-Wave Pairing in One-unit-cell La$_3$Ni$_2$O$_7$ Thin Film Via Substrate-Induced Hole Doping

    cond-mat.supr-con 2025-12 conditional novelty 5.0 of 10

    Hole doping drives the pairing in strained 1-unit-cell La3Ni2O7 films from weak/nonexistent to a d_x2-y2 (then d_xy) wave, through intra-layer spin fluctuations within the γ pocket.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.