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Interlayer interactions in $\text{La}_3\text{Ni}_2\text{O}_7$ under pressure: from $s^{\pm}$ to $d_{xy}$-wave superconductivity

T0 review · 2 major / 3 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Interlayer Coulomb repulsion switches La3Ni2O7's leading pairing from s±-wave to dxy-wave.

desk verdict A clean mRPA calculation of interlayer Coulomb effects in La3Ni2O7, but the spin channel omits an exchange term present in the model Hamiltonian, leaving the central s±→dxy transition unproven for the stated model. read the letter →

arxiv 2502.08425 v3 pith:XIOZPPDQ submitted 2025-02-12 cond-mat.supr-con cond-mat.str-el

Lauro B. Braz , George B. Martins , Luis G. G. V. Dias da Silva This is my paper · ORCID
classification cond-mat.supr-concond-mat.str-el
keywords La3Ni2O7bilayernickelateinterlayerCoulombinteractiondxy-wavepairings±-wavematrixRPAchargesusceptibilitysymmetrytransition
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

The paper asks whether the dominant superconducting pairing symmetry of the pressurized bilayer nickelate La3Ni2O7 survives once interactions between the two NiO2 layers are included. Working with a two-orbital, two-layer tight-binding model and matrix-RPA pairing theory, it answers: interlayer Coulomb repulsion $U_\perp$ suppresses the charge susceptibility at the Fermi-surface nesting vectors, and beyond $U_\perp \approx 0.2U = 0.23$ eV (for $U = 1.16$ eV) the leading gap changes from nodeless $s^\pm$-wave to nodal $d_{xy}$-wave. This matters because earlier RPA calculations for this material kept mainly on-site interactions and predicted $s^\pm$-wave pairing, while neutron, X-ray and RIXS experiments indicate strong interlayer magnetic coupling. If the calculation is right, the gap symmetry of La3Ni2O7 is not a fixed feature but a tunable one that depends sensitively on interlayer interactions.

What carries the argument

The central object is the matrix-RPA pairing interaction $\Gamma(k,k')$, built from RPA-enhanced spin and charge susceptibility matrices, together with the linearized gap eigenvalue equation on the Fermi surface. The mechanism runs through the charge channel: the interlayer interaction matrices $U_\perp, U'_\perp, J_\perp$ appear only in the charge interaction matrix $\hat{U}_c$, so increasing $U_\perp$ reduces the leading eigenvalue of $\hat{\chi}_c(q,0)$ near $\Gamma$ and $M$ while $\hat{\chi}_s$ stays unchanged. Because the charge term enters the pairing vertex with a negative sign, this reduction increases the pairing interaction at the nesting vectors and makes sign-changing solutions, specifically the $d_{xy}$-wave gap obeying $\Delta_\alpha(k) = -\Delta_\alpha(k + q_1)$ and $\Delta_\alpha(k) = -\Delta_\alpha(k + q_2)$, more favorable. Pocket-resolved calculations show that the $\gamma$ pocket (mostly $d_{3z^2-r^2}$) and the $\alpha$ pocket (mostly $d_{x^2-y^2}$) develop enhanced pairing between them as $U_\perp$ grows, so interorbital $\gamma$–$\alpha$ pairing is what selects $d_{xy}$.

What would settle it

Measure the low-temperature London penetration depth or specific heat of pressurized La3Ni2O7: the $d_{xy}$ gap has line nodes, so these quantities should follow power laws rather than the activated exponential behavior of a nodeless $s^\pm$ gap. A direct calculation that includes $J_\perp$ in the spin channel of the same two-orbital model and asks whether the $U_\perp^c \approx 0.2U$ transition survives would settle the modeling assumption.

Watch

Extended reading notes

Core claim

Using the two-orbital bilayer model built from DFT-derived bands, with onsite $U = 1.16$ eV, the matrix-RPA linearized gap equation at $U_\perp = 0$ returns a leading $s^\pm$-wave gap, reproducing earlier results. As $U_\perp/U$ grows, the charge susceptibility matrix is suppressed near the nesting vectors $q_1$ and $q_2$ around the $M$ point, while the spin susceptibility remains unchanged because the interlayer terms appear only in the charge interaction matrix. The charge term enters the pairing vertex with a negative sign, so this suppression strengthens the pairing interaction at those nesting vectors and favors sign-changing solutions with $\Delta_\alpha(k) = -\Delta_\alpha(k + q_i)$. The transition happens at $U_\perp^c \approx 0.2U = 0.23$ eV for $U = 1.16$ eV; beyond it the leading gap is $d_{xy}$-wave, nodal, and built from interorbital pairing across all three Fermi pockets, unlike the nodeless $s^\pm$ gap. The $s^\pm$ solution regains dominance only for larger on-site $U \gtrsim 1.3$ eV or if the interlayer Hund coupling is lowered.

Load-bearing premise

The whole transition rests on the modeling choice that interlayer Coulomb repulsion suppresses only the charge-fluctuation channel while leaving spin fluctuations unchanged; if interlayer exchange also acts in the spin channel, the $d_{xy}$-wave outcome is not guaranteed.

Editorial extensions

If this is right

  • If correct, the leading pairing state is nodal, so low-temperature penetration depth, specific heat, and thermal conductivity in pressurized La3Ni2O7 should show power-law behavior rather than the exponential activation of a nodeless $s^\pm$ gap.
  • The near-degeneracy of the two channels means modest changes in interaction strengths, strain, or doping can flip the gap symmetry; the material sits close to a pairing-symmetry boundary.
  • No single Fermi pocket reproduces the transition, so a credible minimal model must keep all three pockets and both $d_{3z^2-r^2}$ and $d_{x^2-y^2}$ orbitals.
  • The same interlayer-driven mechanism can be expected to operate in other multilayer nickelates with competing pairing channels, such as trilayer La4Ni3O10.

Reading between the lines

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

  • A testable extension the authors do not pursue is mapping the gap nodes directly: quasiparticle interference or angle-resolved specific heat should reveal the nodal structure of the $d_{xy}$ state and distinguish it from $s^\pm$.
  • Because the effect is driven by charge rather than spin fluctuations, a strong-coupling treatment in which interlayer exchange enters the spin channel could shift or erase the transition; comparing such models would test whether the $d_{xy}$ outcome is robust.
  • Strain engineering, already shown to tune interlayer exchange in La3Ni2O7 thin films, offers a way to sweep $U_\perp$ experimentally and look for the symmetry switch under controlled conditions.
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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

2 major / 3 minor

Summary. The paper studies a two-layer, two-orbital itinerant-electron model of La3Ni2O7 under pressure, using the matrix random-phase approximation (mRPA) to compute spin and charge susceptibilities and then solving the linearized gap equation. The central claim is that increasing the interlayer Coulomb repulsion U_perp suppresses the charge susceptibility near the nesting vectors and changes the leading superconducting pairing symmetry from s±-wave to dxy-wave, with a transition around U_perp ≈ 0.2U = 0.23 eV for U = 1.16 eV. The authors present phase diagrams, gap functions on the Fermi surface, and a pocket-by-pocket analysis attributing the transition to enhanced interorbital pairing between the gamma and alpha pockets. The manuscript also includes data-availability and reproducibility statements.

Significance. If the central claim is correct, the paper establishes a concrete microscopic route by which interlayer interactions can reverse the pairing symmetry in a bilayer nickelate, which is a timely and relevant result for the La3Ni2O7 debate. The work uses a standard mRPA framework, provides openly available data, and gives a clear phase diagram. The main significance is therefore conditional: the mechanism rests entirely on the assertion that interlayer interactions enter only the charge channel and leave the spin channel unchanged. If that modeling choice is inconsistent with the stated Hamiltonian, the central result, and not just its numerical value, needs to be re-examined.

major comments (2)
  1. [Eq. (4) and Eq. (6)] The model Hamiltonian in Eq. (4) includes the interlayer spin-exchange term -2J_perp S_{ipA} · S_{iqB}, but the spin interaction matrix in Eq. (6) contains no interlayer entries; the paper states that interlayer interaction terms appear only in the charge interaction matrices. This means the solved model is not the Hamiltonian of Eq. (4), and the difference is directly load-bearing: the mechanism in Sec. IV.B is built on the fact that chi_s is independent of U_perp while chi_c is suppressed. Under the spin-rotational invariance assumed in Sec. II.B, the J_perp term should contribute to U_s, and then chi_s would also evolve with U_perp, changing the competition in Eq. (12). The authors should either derive the omission from a symmetry or approximation explicitly, or include J_perp in U_s and recompute the phase diagram.
  2. [Appendix A and Sec. IV.B] The sensitivity analysis in Appendix A varies J_perp only in the charge matrix U_c, where setting J_perp = 0 shifts the transition from U_perp/U ≈ 0.3 to ≈ 0.5. This does not test the missing contribution of J_perp to the spin matrix U_s, which is precisely the term omitted in Eq. (6). Since the magnitude of J_perp is taken as U_perp/4 and the interlayer exchange is significant in the stated model, the effect of putting J_perp into U_s could be comparable to the entire reported transition shift. The authors must perform this test or explicitly justify that the J_perp spin term is irrelevant in the itinerant weak-coupling regime considered.
minor comments (3)
  1. [Throughout] There are several typographical errors, including 'paring' for 'pairing' in Sec. IV.C and 'bilayer' for 'bilayer' in the Introduction; these should be corrected in a final revision.
  2. [Fig. 4 caption and Sec. IV.A] The text in Fig. 4(a) reads 'U/U_perp = 0' where the context makes clear that the intended ratio is U_perp/U = 0; this should be fixed.
  3. [Sec. IV.A and Fig. 3] The paper uses both 'd_xy-wave' and 'd_{x^2-y^2}-wave' for different symmetry labels in the text and figures; for example, Fig. 4 discusses 'd_{x^2-y^2} waves' while the abstract and Fig. 3 emphasize d_xy. The notation should be made consistent, especially because d_xy and d_{x^2-y^2} are distinct irreps in this lattice context.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the s±→dxy transition is a numerical output of the RPA gap equation, not a fitted or self-referential result.

full rationale

The central claim (interlayer interactions drive a transition from s±-wave to dxy-wave pairing) is obtained by solving the RPA pairing eigenvalue problem, Eq. (14), with λ_max and Δ_max(k) as outputs. The tight-binding parameters come from Ref. [6], U≈1.16 eV is taken from Ref. [12], and U_perp is scanned as a control parameter; the quoted transition U_perp^c≈0.2U=0.23 eV is therefore a computed quantity, not a parameter fitted to the dxy outcome. The only self-citation, Ref. [27], appears in a list of standard mRPA references [23–27], and nothing in the derivation depends on the truth of that citation; the framework is independently established in Refs. [23–25]. The assumption that interlayer interactions enter only the charge susceptibility (Eq. (6), 'Notice that the interlayer interaction terms (U⊥, U′⊥) appear only in the charge interaction matrices') is a stated modeling choice; while one may argue that the interlayer spin exchange in Eq. (4) should also enter Ûs, that is an internal-consistency/correctness issue, not a circular reduction—the calculation solves the equations as written and the dxy result is not equivalent to the input by construction.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on four chosen model parameters (U, J/U, J_perp/U_perp, U_perp) and four background assumptions, of which the charge-only treatment of interlayer interactions is the most consequential. No new entities are introduced.

free parameters (4)
  • Onsite Hubbard U = 1.16 eV
    Reference value taken from Ref. [12], which estimates Tc ≈ 80 K; the phase diagram is computed around this value.
  • Hund's coupling ratio J/U = 0.25
    Set J = U/4, consistent with previous studies [7]; varying it does not change the leading symmetry (Appendix A).
  • Interlayer Hund's ratio J_perp/U_perp = 0.25 (default); 0 in sensitivity check
    Default J_perp = U_perp/4; lowering it shifts the transition from U_perp/U ≈ 0.2 to ≈ 0.5, a quantitative but not qualitative change.
  • Interlayer Coulomb U_perp = scanned; critical U_perp_c ≈ 0.2U = 0.23 eV
    Control parameter of the study; the central claim is the transition as U_perp increases.
assumptions (4)
  • domain assumption The two-layer, two-orbital tight-binding Hamiltonian of Luo et al. [6] accurately describes the low-energy fermiology of La3Ni2O7 under pressure.
    All band structure and Fermi surface inputs come from this model; no direct experimental verification of the model in this paper.
  • domain assumption Matrix-RPA is valid for U ≈ 1.16 eV, where U/|t_z_perp| ≈ 1.83, and the Stoner criterion is not satisfied for the studied parameter range.
    RPA is an uncontrolled approximation at intermediate coupling; the authors verify the Stoner criterion but not convergence against exact methods.
  • ad hoc to paper Interlayer Coulomb interactions affect pairing only through the charge susceptibility; the spin susceptibility is independent of U_perp and J_perp.
    Given in Eq. (6): U_perp terms appear only in Ûc, not Ûs. This separation is the core of the proposed mechanism.
  • ad hoc to paper Kanamori-type relations U' = U - 2J and U'_perp = U_perp - 2 J_perp with J = U/4, J_perp = U_perp/4.
    Standard local rotational invariance, but the 1/4 ratio is a choice; sensitivity to it is tested in Appendix A.

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Pith. "Pith review of Interlayer interactions in $\text{La}_3\text{Ni}_2\text{O}_7$ under pressure: from $s^{\pm}$ to $d_{xy}$-wave superconductivity." pith.science (2026). https://pith.science/paper/XIOZPPDQ

@misc{pith2026250208425,
  author       = {Pith},
  title        = {Pith review of: Interlayer interactions in $\textLa_3\textNi_2\textO_7$ under pressure: from $s^\pm$ to $d_xy$-wave superconductivity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XIOZPPDQ}},
  note         = {Machine review of arXiv:2502.08425}
}
abstract

We investigate the role of \emph{interlayer} interaction terms in the competition between different superconducting gap symmetries in the bilayer nickelate $\text{La}_3\text{Ni}_2\text{O}_7$ under high pressure. We study a two-layer, two-orbital electron model that encompasses both intra- and interlayer Coulomb interaction terms within the matrix random-phase approximation. We find that interlayer interactions favor a $d_{xy}$-wave superconducting pairing symmetry over the $s^{\pm}$-wave symmetry, which has been found to prevail when interlayer interactions are disregarded. Moreover, our findings indicate that interlayer interactions enhance the interorbital pairing, incorporating contributions from all three electron pockets, arising from both $d_{3z^2-r^2}$ and $d_{x^2-y^2}$ orbital character, resulting in nodes within the gap function (not present in the $s^{\pm}$-wave state) and consequently favoring the $d_{xy}$-wave pairing.

Figures

Figures reproduced from arXiv: 2502.08425 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic representation of the bilayer La [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Contour plots for the main eigenvalue of the spin (a) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Phase diagram showing the log of the ratio be [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Susceptibilities and the sign changes in the super [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Enhancement of the pairing vertex function [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Color maps of the leading superconducting gap func [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Leading gap functions ∆ [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.