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Orbital-selective correlation effects and superconducting pairing symmetry in a multiorbital $t$-$J$ model for bilayer nickelates

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A bilayer two-orbital $t$-$J$ model for La$_3$Ni$_2$O$_7$ yields either extended $s$-wave or $d_{x^2-y^2}$-wave pairing, with the dominant orbital switching as the $z^2$ bonding band moves through the Fermi level.

desk verdict A plausible mean-field map of s- and d-wave pairing in bilayer nickelates, with a real internal-consistency question at the crossing that drives the film/bulk story. read the letter →

arxiv 2502.09195 v1 pith:IUKIIQMI submitted 2025-02-13 cond-mat.supr-con cond-mat.str-el

Guijing Duan , Zhiguang Liao , Lei Chen , Yiming Wang , Rong Yu , Qimiao Si This is my paper · ORCID
classification cond-mat.supr-concond-mat.str-el
keywords bilayernickelatesLa3Ni2O7orbital-selectiveMottphysicst-Jmodelsuperconductingpairingsymmetryextendeds-waved-waveslave-spinmethod
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 superconductivity in the bilayer nickelate La$_3$Ni$_2$O$_7$ is controlled by orbital-selective correlations, captured by an effective bilayer two-orbital $t$-$J$ model. It shows that the leading superconducting pairing is either extended $s$-wave ($A_{1g}$), carried by interlayer pairing in the $z^2$ or $x^2-y^2$ orbital, or $d_{x^2-y^2}$-wave ($B_{1g}$), carried by in-plane pairing in the $x^2-y^2$ orbital. The paper further shows that shifting the $z^2$ bonding band through the Fermi level switches the dominant pairing orbital, which links bulk crystals under pressure (likely $s$-wave) to strained thin films with an expanded $c$-axis (likely $d$-wave and a lower $T_c$).

What carries the argument

The machinery is the slave-spin renormalized bilayer two-orbital $t$-$J$ Hamiltonian (Eq. 1): quasiparticle weights $Z_\alpha$ taken from a prior slave-spin calculation renormalize the hopping, and orbital-diagonal intralayer and interlayer exchange couplings $J_{\parallel\alpha\alpha}$ and $J_{\perp\alpha\alpha}$ generate pairing through a Bogoliubov-Hubbard-Stratonovich decomposition of the spin-singlet channel. Pairing amplitudes $\Delta_{\delta\alpha}$ are solved self-consistently, and superpositions of gap functions are classified by irreducible representations of $D_{4h}$. The quantity that carries the orbital-switch argument is $\Delta E$, the energy of the $z^2$ bonding band top relative to the Fermi level.

What would settle it

A direct falsifier: measure the momentum dependence of the superconducting gap on bulk La$_3$Ni$_2$O$_7$ under pressure; a nodeless fully gapped Fermi surface supports the $s$-wave scenario, while nodes along $k_x = \pm k_y$ support the $d$-wave scenario. Additionally, if ARPES or quantum oscillations show that the $z^2$ bonding band top stays below the Fermi level while the 80 K superconducting state persists, the claimed $s_z$ pairing mechanism would be contradicted.

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Extended reading notes

Core claim

The central result is a pairing phase diagram in the space of interlayer exchange couplings $J_{\perp zz}$ and $J_{\perp xx}$ (normalized by intralayer $J_{\parallel xx}$) for the bilayer two-orbital $t$-$J$ model. Three leading channels compete: a $d_{x^2-y^2}$-wave ($B_{1g}$) state with in-plane pairing in the $x^2-y^2$ orbital, an extended $s$-wave ($A_{1g}$) state with interlayer pairing in the $x^2-y^2$ orbital ($s_x$), and an extended $s$-wave state with interlayer pairing in the $z^2$ orbital ($s_z$). Increasing either interlayer exchange coupling favors $s$-wave pairing, and within the $s$-wave region the dominant orbital crosses over from $z^2$ to $x^2-y^2$. When the top of the $z^2$ bonding band drops below the Fermi level, the leading pairing can switch from $s_z$ to $d_x$ (or to $s_x$), an effect that the authors identify with the relationship between $c$-axis strain and superconductivity in bulk crystals versus thin films.

Load-bearing premise

The calculation assumes that the slave-spin renormalized low-energy model, with $Z_\alpha$ values from Ref. [21] and with only orbital-diagonal intralayer and interlayer exchange couplings, faithfully represents La$_3$Ni$_2$O$_7$ near an orbital-selective Mott phase.

Editorial extensions

If this is right

  • For bulk La$_3$Ni$_2$O$_7$ under pressure, where the $z^2$ bonding band crosses the Fermi level, the model places the leading pairing in the $s_z$ extended-$s$-wave channel with a fully gapped, nodeless Fermi surface.
  • For strained thin films with an expanded $c$-axis, the $z^2$ band top can fall below the Fermi level, switching the leading channel to $d_{x^2-y^2}$-wave pairing with nodes; the reduced pairing amplitude offers an explanation for the lower $T_c$ observed in films.
  • In the $s$-wave region, the in-plane and interlayer gap components typically have opposite signs, which stabilizes the state without nodes; gap anisotropy can look similar for $d_x$ and $s_z$, so phase-sensitive measurements are needed to distinguish them.
  • Hole doping, as inferred in La$_2$PrNi$_2$O$_7$ thin films, can counteract the strain-induced band shift and may keep the pairing symmetry the same as in bulk.

Reading between the lines

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

  • If the pairing symmetry really is set by the $z^2$ band-top position, then hydrostatic pressure experiments that continuously tune $c$ should show a pairing-symmetry crossover at the pressure where the band top crosses the Fermi level, not merely a $T_c$ maximum.
  • The $s+id$ state near the $s$/$d$ boundary, although nearly degenerate with pure $d$-wave in this calculation, would be a rare spontaneous time-reversal-symmetry-breaking superconductor; a zero-field muon spin rotation measurement could look for it if a sample is tuned to the boundary.
  • The model's dependence on only diagonal exchange couplings suggests that including inter-orbital (Hund's) exchange terms could shift the crossover boundaries and is a natural next step.
  • Comparing gap structures on the $z^2$ and $x^2-y^2$ Fermi pockets in the same material, e.g. by ARPES or scanning tunneling spectroscopy, would directly test the orbital dichotomy claimed here.
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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 manuscript studies superconducting pairing in a bilayer two-orbital t-J model for La3Ni2O7, constructed from a slave-spin renormalized bilayer two-orbital Hubbard model near an orbital-selective Mott phase. Using a self-consistent mean-field decomposition of the exchange couplings, the authors obtain a phase diagram with three competing leading channels: an in-plane d_{x^2-y^2}-wave (B1g) gap in the x^2-y^2 orbital, and two extended s-wave (A1g) gaps carried by interlayer pairing in either the z^2 or the x^2-y^2 orbital. They further show that moving the z^2 bonding band top across the Fermi level, by varying the crystal-field splitting, can switch the leading pairing channel from z^2-dominated s-wave to x^2-y^2-dominated d-wave, and they use this to rationalize a possible s-wave bulk state versus a d-wave, reduced-Tc thin-film state.

Significance. If its central prediction survives closer scrutiny, the paper gives a concrete and falsifiable route from orbital-selective correlations to pairing symmetry: bulk La3Ni2O7 would be s-wave, while c-axis-expanded thin films could be d-wave with a reduced Tc. The mean-field machinery is standard, the self-consistent gap equations are clearly stated, and the symmetry classification in the Supplemental Material is a useful reference. The paper also makes explicit predictions for gap anisotropy and for phase-sensitive measurements that could distinguish the competing states. The significance is tempered, however, by the fact that the key crossover is computed with the slave-spin quasiparticle weights frozen at values taken from the authors' companion calculation, and by the omission of off-diagonal exchange channels without a quantitative justification.

major comments (3)
  1. [Model and method, Eq. (1); Fig. 4] The central sz-to-dx pairing crossing is computed by varying the crystal-field splitting Δε while the slave-spin quasiparticle weights Zα are held fixed at the values taken from Ref. [21]. In the slave-spin formalism, Zα renormalizes the kinetic energy and depends on the orbital occupancy; as the paper itself notes, increasing Δε hole-dopes the z^2 orbital and electron-dopes the x^2-y^2 orbital. The calculation therefore moves the very band whose position is claimed to drive the orbital-selective pairing change while freezing the orbital-selective renormalization. To establish the crossing, the authors should recompute Zα, and ideally the effective exchange couplings, at each Δε along the trajectory of Fig. 4, or at minimum provide a sensitivity analysis showing that the crossing and the film/bulk conclusion are robust to realistic changes in Zα.
  2. [Eq. (1) and Fig. 1(b)] The exchange interactions are restricted to orbital-diagonal channels Jδαα. In a multiorbital t-J model derived from a Hubbard Hamiltonian with finite Hund's coupling, off-diagonal exchange terms, including interorbital spin exchange and pair-hopping (η) terms, are generically present at the same order in t/U. The paper justifies the restriction only by calling these the 'leading exchange interactions' without quantifying them. Since the relative strengths of J⊥zz and J⊥xx determine the sx-versus-sz competition, the neglect of off-diagonal channels should be justified from the microscopic parameters, or their possible effect on the phase boundaries should be assessed.
  3. [Model and method and Discussion] The numerical values of Zz and Zx used in the calculation are never reported, and the estimated ranges for the interlayer exchange ratios (J⊥zz/J∥xx∼1-3, J⊥xx<J⊥zz) are given only verbally. This makes it difficult to reproduce the phase diagram or to judge how close the physically estimated point lies to the dx/sx/sz boundaries. The authors should state the actual Zα values, the Δε values used in Fig. 4, and show how the phase boundaries shift when these inputs are varied within a plausible range.
minor comments (4)
  1. [Abstract and main text] The abstract uses 'extended s-wave' while the text uses 'extensive s-wave' in several places; the terminology should be made consistent.
  2. [Fig. 4 and Discussion] The U and JH parameters used in Fig. 4 are not stated in the caption; the reader must infer that they are the same as in Fig. 2.
  3. [Various] There are several typographical errors, including 'ansiotropic' in the main text, 'hole doing' in the Fig. 1 caption, and 'CROSS BETWEEN THE sx AND sz P AIRING' in the Supplemental Material heading.
  4. [Supplemental Material, Fig. S2] The statement that the sx-sz crossover 'exhibits a crossover' is redundant, and the caption contains 'paring' for 'pairing'; these should be corrected.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction; central pairing phase diagram is independently solved, though the orbital-selective inputs come from the authors' earlier slave-spin work (Ref. [21]).

full rationale

The pairing calculation is self-contained once the model in Eq. (1) and the exchange couplings are specified: the gap functions are solved self-consistently through the Hubbard-Stratonovich decoupling of Eq. (2), and the phase diagram in Fig. 2 is computed rather than fitted to any measured Tc or gap amplitude. No equation defines the predicted pairing symmetry in terms of an input, and the Delta E-driven sz to dx crossing in Fig. 4 follows from the change in Fermi-surface topology when the z2 bonding band passes through the chemical potential, not from a fitted constraint. The circularity-adjacent feature is that the kinetic renormalization factors Z_alpha and the estimated J-ratio range are taken from the same group's earlier slave-spin paper (Ref. [21]; 'We take the Z_alpha values ... from the calculation in Ref. [21]'), so the orbital-selective premise is partly self-referential. However, the slave-spin result of Ref. [21] is a separate calculation from the present mean-field pairing solution, and the central pairing phase diagram stands on its own self-consistent solution. A genuine internal-consistency caveat is that Fig. 4 varies the crystal-field splitting while holding Z_alpha fixed, although the text itself notes that increasing Delta_epsilon 'raises the onsite energy of the z2 orbital, corresponding to hole doping the z2 bonding band while electron doping the x2-y2 orbital'; in the slave-spin formalism the quasiparticle weights would in principle shift with orbital occupancy. This is a limitation of the low-energy model, not a circular derivation. The paper's own caveat that 'precise determination of the values awaits future experimental and theoretical studies' is a normal parameter-uncertainty statement. Conclusion: no significant circularity; score 2 reflects the same-group parameterization of the model inputs.

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

The central claim is a property of an effective model, and the model itself rests on inputs imported from earlier work by the same group (slave-spin weights Zα and the orbital-selective Mott proximity from Ref. [21]). The exchange couplings J⊥xx and J⊥zz are scanned rather than computed, and the Δε values in Fig. 4 are chosen to represent bulk and film-like band positions. These are the main places where the paper draws on prior assumptions rather than adding independent evidence.

free parameters (5)
  • Interlayer exchange ratio J⊥zz/J∥xx = scanned 0 to 4 (dimensionless)
    The main phase diagram Fig. 2(a) is plotted against this ratio; the value in real material is only estimated as 1-3 from Ref. [21], so any quantitative pairing symmetry assignment depends on this undetermined input.
  • Interlayer exchange ratio J⊥xx/J∥xx = scanned 0 to 4 (dimensionless)
    Controls competition between sx, sz, and dx channels; treated as a free parameter range because 'J⊥xx can be mediated through strong Hund's coupling' is not quantitatively fixed.
  • Crystal-field splitting Δε = εz' - εx' = -0.239 eV to -0.359 eV (varied)
    The key tuning knob in Fig. 4; changing it moves the z2 bonding band top through the Fermi level and triggers the pairing crossing. The values are chosen to represent bulk and film-like band positions, not derived.
  • Hubbard U and Hund's ratio JH/U = U = 4.8 eV, JH/U = 0.2
    Picked as a representative strongly correlated parameter set from Ref. [21]; it determines Zα and J∥ values used throughout.
  • Slave-spin quasiparticle weights Zα = not given in this paper (from Ref. [21])
    Kinetic energy renormalization in Eq. (1) is imported from Ref. [21]; the central results inherit these numbers without re-derivation.
assumptions (5)
  • domain assumption Slave-spin mean-field theory correctly captures the low-energy renormalized band structure and exchange couplings of the bilayer two-orbital Hubbard model.
    Used to justify Eq. (1); no independent benchmark is provided in this paper. Refs. [67-69] are cited.
  • domain assumption The low-energy sector is faithfully represented by a two-orbital t-J model with only orbital-diagonal intralayer and interlayer exchange couplings.
    The paper keeps only J∥αα and J⊥αα (nearest neighbor, Fig. 1b); off-diagonal exchange, pair hopping, and density-density J terms are dropped without quantitative justification.
  • ad hoc to paper The orbital-selective Mott proximity scenario of Ref. [21] applies to La3Ni2O7 at N = 3 electrons per unit cell.
    Anchors the whole model; if the real material is not near an orbital-selective Mott phase, the derived t-J model and its pairing conclusions are not applicable.
  • ad hoc to paper Interlayer exchange in the x2-y2 orbital, J⊥xx, can be sizable via Hund's coupling.
    Ref. [61] is cited for this mechanism; the magnitude is not computed here and is instead scanned over a wide range.
  • domain assumption Bogoliubov Hubbard-Stratonovich mean-field decoupling in the spin-singlet channel captures the ground-state pairing of the t-J model.
    Standard method, but it omits fluctuation effects, competing orders, and triplet channels; the phase boundaries in Fig. 2(a) are mean-field estimates.

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Pith. "Pith review of Orbital-selective correlation effects and superconducting pairing symmetry in a multiorbital $t$-$J$ model for bilayer nickelates." pith.science (2026). https://pith.science/paper/IUKIIQMI

@misc{pith2026250209195,
  author       = {Pith},
  title        = {Pith review of: Orbital-selective correlation effects and superconducting pairing symmetry in a multiorbital $t$-$J$ model for bilayer nickelates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IUKIIQMI}},
  note         = {Machine review of arXiv:2502.09195}
}
abstract

The recent discovery of superconductivity in La$_3$Ni$_2$O$_7$ raises key questions about its mechanism and the nature of pairing symmetry. This system is believed to be described by a bilayer two-orbital Hubbard model. The considerations of orbital-selective Mott correlations motivate a bilayer two-orbital $t$-$J$ model and, accordingly, we study the superconducting pairing in this model. We obtain an overall phase diagram of superconductivity, where the leading channel has either extended $s$-wave or $d_{x^2-y^2}$-wave symmetry. Our analysis highlights how the orbital-selective correlations affect the superconducting pairing via the interlayer exchange couplings and low-energy electronic structure. In particular, we find that the dominant orbital for the pairing may change between $z^2$ and $x^2-y^2$ when the position of the bonding $z^2$ band is varied by tuning either the $c$-axis lattice constant or electron concentration strength. We discuss the implications of these results for the superconductivity in both bulk La$_{3}$Ni$_{2}$O$_{7}$ and its thin film counterpart.

Figures

Figures reproduced from arXiv: 2502.09195 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Global phase diagram of orbital-selective [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Superconducting phase diagram with [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Superconducting gaps projected onto the Fermi [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. (a) Band structures with different crystal splitting [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 5 Pith papers

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

  1. Magnetic Order in bilayer Ruddlesden-Popper Nickelates

    cond-mat.supr-con 2026-07 conditional novelty 6.0 of 10

    Combining superexchange with RKKY interactions between orbital-selective local moments reproduces the (π/2,π/2) magnetic order and ~80 meV spin excitations of bilayer nickelate La₃Ni₂O₇.

  2. Possible Enhancement of Superconductivity in Ambient-Pressure La$_3$Ni$_2$O$_7$ Thin Film

    cond-mat.supr-con 2026-03 conditional novelty 6.0 of 10

    A δ pocket of dz2 character near Γ enhances s± pairing in a model of La3Ni2O7 films by nesting with the γ pocket, raising the Eliashberg eigenvalue λ.

  3. Role of correlations in Ruddlesden-Popper bilayer nickelates under compressive strain

    cond-mat.str-el 2025-08 conditional novelty 6.0 of 10

    At -2% compressive strain, dynamic electron correlations make a dz2-derived flat band cross the Fermi level in bilayer La3Ni2O7, creating an extra Fermi pocket absent in static DFT+U; at -3% the pocket disappears.

  4. Evolution from intralayer to interlayer superconductivity in a bilayer $t$-$J$ model

    cond-mat.str-el 2025-07 conditional novelty 6.0 of 10

    In a bilayer t-J-J⊥ ladder, increasing the interlayer spin coupling J⊥ drives a transition from intralayer to interlayer superconductivity, with a charge-density-wave intermediate phase at low doping and enhanced inte...

  5. 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.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.