{"id":"8938234d-d2e1-4c56-846e-015d6be2a592","arxiv_id":"2502.09195","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"In a bilayer two-orbital t-J model for La3Ni2O7, the leading pairing is either extended s-wave or d_{x^2-y^2}-wave, and moving the z2 bonding band through the Fermi level switches the dominant pairing orbital from z2 to x2-y2.","lead":"A theory paper maps which superconducting state a bilayer nickelate model favors, finding either an s-wave or d-wave gap depending on magnetic couplings and the position of a key orbital band. The result suggests bulk and thin-film nickelates may pair in different symmetries, which could explain their very different transition temperatures.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The predicted sz-to-dx pairing crossing is computed at fixed slave-spin weights Zα, so the central orbital-selective crossover is not yet shown to be self-consistent.","rationale":"The reader's weakest assumption concerns the fidelity of the imported Zα values and the restriction to diagonal exchange couplings. My read is consistent with that: the quantitative mapping to La3Ni2O7 depends on the slave-spin inputs. But the more specific load-bearing issue is that the central novel result, Fig. 4, varies the crystal field that controls the z2 band position while holding Zα and J⊥ fixed; since that variation changes orbital occupancies, the slave-spin renormalization should be recalculated along the scan. This does not impugn the authors' earlier slave-spin calculation or the OSMP picture; it asks for the model to be solved self-consistently at the same level of approximation used to define it. The calculation as presented is internally coherent as a mean-field study of a fixed low-energy Hamiltonian, and the phase diagrams are plausible. The concrete test would determine whether the crossing is an artifact of the frozen Zα assumption. Because this is a checkable refinement rather than a demonstrated error, the reader's CONDITIONAL verdict remains appropriate, and I would not move it to ACCEPT or REJECT. I therefore mark the verdict as UNCHANGED and note partial agreement with the reader's weakest assumption.","tokens_in":11449,"tokens_out":11308,"duration_ms":109728,"concrete_test":"Run a slave-spin (or slave-boson) self-consistent calculation for the bilayer two-orbital Hubbard model at fixed total occupancy N = 3, sweeping Δε over the range used in Fig. 4 (from about −0.359 eV to −0.239 eV), and compute Z_z(Δε), Z_x(Δε), and the renormalized interlayer exchange couplings. Then re-solve the Bogoliubov–de Gennes gap equations of Eq. (1) with these Δε-dependent parameters. If the leading pairing amplitude still shows the sz-to-dx crossing as ΔE becomes negative, the central prediction is robust; if the crossing is shifted, weakened, or absent, the paper should either present the self-consistent phase diagram or limit the prediction to the fixed-Zα parameterization. For the doping part of the abstract, repeat the same procedure at N = 3 − δ (e.g., δ = 0.2 per Ni) and check whether the crossing occurs as a function of δ.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that moving the z2 bonding band relative to the Fermi level—by c-axis strain or by carrier doping—switches the leading pairing from sz (interlayer z2 s-wave) to dx (in-plane x2-y2 d-wave), which underlies the film-vs-bulk prediction. This is computed in Fig. 4 by varying the crystal-field splitting Δε = ε'_z - ε'_x, while Zα is fixed at the values taken from Ref. [21]. But changing Δε redistributes charge between orbitals, as the paper itself states ('increasing Δε 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 Zα depend on orbital occupancy, so they should be re-evaluated at each Δε. The same applies to the exchange couplings if they are derived from the correlated solution. By freezing Zα, the calculation holds the orbital-selective renormalization fixed while moving the very band whose position is claimed to drive the orbital-selective pairing change. If Z_z and the effective J's change as the z2 band sinks below the Fermi level, the sz pairing amplitude and the crossing location could be substantially modified, and the film/bulk conclusion would need revision. This is an internal consistency issue of the low-energy model, not a challenge to the orbital-selective Mott proximity itself.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11773,"tokens_out":5544,"duration_ms":56938,"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":[{"comment":"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.","section":"Model and method, Eq. (1); Fig. 4"},{"comment":"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.","section":"Eq. (1) and Fig. 1(b)"},{"comment":"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.","section":"Model and method and Discussion"}],"minor_comments":[{"comment":"The abstract uses 'extended s-wave' while the text uses 'extensive s-wave' in several places; the terminology should be made consistent.","section":"Abstract and main text"},{"comment":"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.","section":"Fig. 4 and Discussion"},{"comment":"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.","section":"Various"},{"comment":"The statement that the sx-sz crossover 'exhibits a crossover' is redundant, and the caption contains 'paring' for 'pairing'; these should be corrected.","section":"Supplemental Material, Fig. S2"}],"recommendation":"major_revision","confidential_remarks":"The paper's load-bearing inputs, Zα and the J-ratio estimates, come from the authors' own arXiv preprint Ref. [21], which is not yet peer-reviewed to my knowledge. The present manuscript would be considerably stronger if it either made that companion calculation's results more explicit or reduced its sensitivity to them. My major comments are directed at verifiability of this dependence rather than at the overall approach."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the paper in two lines: a slave-spin mean-field study of a bilayer two-orbital t-J model for La3Ni2O7, producing a pairing phase diagram where the leading channel is either extended s-wave or d_{x^2-y^2}-wave, and where the dominant orbital switches from z^2 to x^2-y^2 as the z^2 bonding band moves below the Fermi level. The genuinely new content is the J⊥xx-J⊥zz phase diagram of Fig. 2(a) and the ΔE-driven crossing of Fig. 4; the framework comes from the same group's earlier Refs. 13 and 21.\n\nWhat's done well: the mean-field decoupling and self-consistent gap equations are standard, the phase diagram is internally consistent as a mean-field result, and the gap-anisotropy discussion is careful. The authors are explicit that distinguishing dx from sz requires phase-sensitive measurements, and they don't overclaim their J estimates. That's honest.\n\nThe soft spots, in order: first, the central crossing calculation fixes the slave-spin quasiparticle weights Zα at values taken from Ref. [21] while varying Δε, which changes orbital occupancies. In slave-spin theory, Zα depends on those occupancies, so the calculation moves the z^2 band without letting the orbital-selective renormalization respond. The paper itself notes that increasing Δε holes the z^2 band and electron-dopes x^2-y^2, so this is not a small perturbation. The crossing location, and possibly its existence, could shift if Z_z and Z_x were re-computed at each Δε. This is an internal consistency issue of the low-energy model, not a challenge to the orbital-selective Mott premise. The authors should state the Zα values used and test sensitivity, or solve the slave-spin problem at each Δε.\n\nSecond, the exchange interactions are limited to orbital-diagonal channels; off-diagonal terms are dropped without quantitative justification. Third, the film/bulk interpretation is plausible but relies on identifying Δε with the c-axis lattice constant and then mapping the crossing to the lower Tc in films. That's an inference, not a direct calculation, and the paper is transparent about it. These are real limitations, not fatal ones.\n\nWho it's for: anyone working on nickelate superconductivity, particularly the thin-film experiments. It deserves peer review; the referee should ask for the Zα self-consistency check and sensitivity analysis. If those hold up, the paper will be a solid contribution.","headline":"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.","tokens_in":12348,"tokens_out":2913,"would_cite":true,"duration_ms":25856,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["bilayer nickelates","La3Ni2O7","orbital-selective Mott physics","t-J model","superconducting pairing symmetry","extended s-wave","d-wave","slave-spin method"],"falsifier":"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.","tokens_in":11206,"feed_emoji":"🔬","tokens_out":5093,"duration_ms":43130,"temperature":0.7,"pith_summary":"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 $T_c$).","feed_headline":"One band position sets nickelate pairing symmetry","feed_subtitle":"A model links nickelate pairing symmetry to the z2 bonding band position, explaining bulk versus thin-film Tc.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the $Z_\\alpha$ quasiparticle weights and the orbital-selective Mott phase diagram the model starts from.","marker":"[21]"},{"why":"Introduces the bilayer two-orbital $t$-$J$ description and the pairing channel classification used here.","marker":"[13]"},{"why":"Provides the self-consistent Bogoliubov/Hubbard-Stratonovich treatment of pairing, reused for the gap equations.","marker":"[71]"},{"why":"Motivates including the interlayer exchange in the $x^2-y^2$ orbital $J_{\\perp xx}$ via Hund's coupling.","marker":"[61]"},{"why":"Reports thin-film superconductivity with a reduced $T_c$, the experimental target for the $d$-wave scenario.","marker":"[24]"},{"why":"Establishes the restriction of the low-energy sector to $z^2$ and $x^2-y^2$ orbitals on which the model is built.","marker":"[12]"}],"fun_headline_variants":["z2 band position flips nickelate pairing symmetry","Bonding z2 band dictates nickelate pairing channel","Orbital-selective correlations set nickelate pairing","How z2 band movement changes nickelate pairing symmetry","Tuning z2 band controls nickelate pairing symmetry"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["z2 band position flips nickelate pairing symmetry","Bonding z2 band dictates nickelate pairing channel","Orbital-selective correlations set nickelate pairing","How z2 band movement changes nickelate pairing symmetry","Tuning z2 band controls nickelate pairing symmetry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000929,"raw_usage":{"total_tokens":4018,"prompt_tokens":1027,"completion_tokens":2991,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":643,"completion_tokens_details":{"reasoning_tokens":2928}},"tokens_in":643,"tokens_out":2991,"duration_ms":20539,"temperature":1.0,"reasoning_tokens":2928,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T22:19:49.248809+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the self-consistent Bogoliubov/Hubbard-Stratonovich treatment of pairing, reused for the gap equations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Motivates including the interlayer exchange in the $x^2-y^2$ orbital $J_{\\perp xx}$ via Hund's coupling."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports thin-film superconductivity with a reduced $T_c$, the experimental target for the $d$-wave scenario."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the restriction of the low-energy sector to $z^2$ and $x^2-y^2$ orbitals on which the model is built."}],"review_version":1}