Variational Quantum Monte Carlo investigations of the superconducting pairing in La$_3$Ni$_2$O$_7$

Da Wang; Qiang-Hua Wang; Yi-Qun Liu

REVIEW

Variational Quantum Monte Carlo investigations of the superconducting pairing in La$_3$Ni$_2$O$_7$

Yi-Qun Liu, Da Wang, Qiang-Hua Wang

Superconductivity · Submitted 12 May 2025 · 74 references

Pith's verdict: First VQMC comparison of s± and d-wave pairing in the bilayer two-orbital nickelate model, but the d-wave ansatz is under-flexible and the tiny energy differences come without error bars; the s± conclusion is plausible but not airtight.

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arXiv ID2505.07341 (version 1)

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CitationDa Wang, Qiang-Hua Wang, Yi-Qun Liu. "Variational Quantum Monte Carlo investigations of the superconducting pairing in La$_3$Ni$_2$O$_7$." arXiv:2505.07341 (2025). https://pith.science/paper/T3CWQVYY

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Editor's note

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I'll give you the short version first: this is a legitimate VQMC study of the bilayer two-orbital model for La3Ni2O7, and it finds that s±-wave pairing wins over d_x2-y2. The new bit is the method—nobody has done a non-perturbative variational comparison for this model—and the result is consistent with the authors' earlier FRG/RMFT work. But the comparison is not as clean as the abstract suggests. The d-wave trial state contains only planar nearest-neighbor pairing, while the s± state has onsite, vertical, and planar channels. So the d-wave energy is an upper bound from a strictly smaller variational manifold. The authors do not test d_xy, longer-range d-wave form factors, or inter-orbital pairing, and they give no error bars on energy differences that are of order 10^-4 to 10^-3 eV. That is a real gap between the data and the claim that s± 'always wins.'

What the paper does well: the VQMC implementation is standard (stochastic reconﬁguration, Jastrow factors, a ﬁnite-size check up to L=12 at one parameter set), and the pairing-amplitude analysis identiﬁes the vertical dz2 bond as the dominant channel, which is physically consistent with the model's largest superexchange. The double-occupancy result—ﬁnite even at large U because of the 1.5 ﬁlling—is a useful observation that strengthens the itinerant picture. The citation pattern is fine: they build on their own prior work and on Luo et al.'s tight-binding parameters, and they don't oversell earlier results.

The soft spots, in order of importance: (1) the asymmetric d-wave ansatz. I cannot rule out that a more ﬂexible d-wave state (with NNN bonds or inter-orbital pairing) would lower the d-wave energy below s±. The paper needs either to extend the d-wave ansatz or to argue convincingly why those channels are irrelevant. (2) The missing statistical error bars. In VQMC, energy differences of a few meV without error bars leave the statistical significance unclear. (3) The comparison is only against d_x2-y2. If the real competitor is d_xy, the paper doesn't address it—and that is not a strawman, given Ref. [54]. (4) The ﬁnite-size check is at a single parameter point; I would like to see it at the edges of the parameter range.

Who is this for? People working on nickelate superconductivity who want a non-perturbative data point on pairing symmetry. It is a respectable paper that will draw attention. I would not desk reject it; a serious referee could push the authors to address the d-wave ﬂexibility issue and add error bars. That would turn a suggestive claim into a solid one.

Referee report

Major objections (3)

[Sec. III, Eq. (7), Figs. 3-4] The central energy comparison is between two variational manifolds of unequal size: the s± trial state contains six pairing channels (z0, x0, z1, x1, z2, x2), while the d_{x^2-y^2} trial state contains only the planar nearest-neighbor channels x2 and z2. Since VQMC minimizes the energy within each manifold, the lower s± energy could reflect the additional variational freedom of the s± ansatz rather than a physical ordering of pairing symmetries. Symmetry does not forbid additional B1g channels in the d-wave trial state, such as longer-range in-plane d-wave form factors or inter-orbital pairing terms. To support the abstract's claim that d_{x^2-y^2} is generically disfavored, the d-wave ansatz should be enlarged to comparable flexibility, or the comparison should be performed with a common set of pairing channels modulated by the two different symmetry form factors.
[Sec. III, Figs. 3-5] No statistical error bars are reported for the optimized energies, and several energy differences in Figs. 3(b) and 4(b) are of order 10^-4 eV, which is small relative to typical VMC statistical fluctuations on a 6x6x2 lattice. The statement that s± 'always wins' requires either error bars on the energy difference or a demonstration that the difference is statistically significant across the entire parameter grid. The finite-size check in Fig. 5 also shows absolute energies without uncertainties, so it does not currently bound the error on the energy difference.
[Sec. II, Eq. (3)] The effective t-J model neglects inter-orbital super-exchange, with the justification that t^{a\bar a}_{ij} and products t^{aa}_{ij}t^{\bar a \bar a}_{ji} are small. This is not quantitatively supported by the tight-binding parameters of Ref. [17] quoted in Sec. II: for the planar nearest-neighbor bond, t^{xz}=0.239 eV is comparable to t^{xx}=-0.483 eV and larger than |t^{zz}|=0.110 eV. The associated planar inter-orbital exchange, of order 4(t^{xz})^2/U ~ 0.03 eV, is larger than the retained z-z planar super-exchange, of order 4(t^{zz})^2/U ~ 0.007 eV. Since the t-J results are used for the 's± always wins' claim, the omission needs a quantitative justification or the t-J conclusions need to be qualified.

Minor objections (4)

[Sec. II] The phrase 'U /greaterorsimilar U′' appears to be a rendering error; it should read U ≳ U′.
[Introduction] The phrase 'seeding light on the itinerant versus local moment picture' should be 'shedding light'; there is also a spacing artifact in 'supe rconducting' in the header.
[Fig. 4 caption] The panel references in the caption are inconsistent: the pairing parameters versus U and J_H are described as panels (e) and (f), while panel (f) is also used for the double-occupancy curves. Please renumber the panels or correct the caption.
[Appendix B, Fig. 5] The vertical axis label says E(eV) but it is not clear whether these are total energies or energies relative to the normal state; please specify this in the caption.

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The paper

The paper claims

The paper claims that strong local correlations in La3Ni2O7 favor s±-wave pairing, led by the 3d_{3z^2-r^2} vertical bond.

The reading

The paper asks which superconducting pairing symmetry wins in the pressurized nickelate La3Ni2O7, a bilayer material with two active orbitals per nickel. Using variational quantum Monte Carlo on the bilayer two-orbital Hubbard model and its strong-coupling extended t-J version, it claims that the s±-wave state—a gap that changes sign between different Fermi pockets—is lower in energy than the d_{$x^{2}$-$y^{2}$}-wave state whenever Hund's coupling is not too weak, and in the t-J model always. The leading pairing component sits on the vertical bond between 3d_{$3z^{2}$-$r^{2}$} orbitals, identifying that orbital as the main actor. If the claim is right, theory and experiment should focus on sign-changing gap states and on interlayer coupling, not on d-wave pairing.

What carries the argument

The central object is the variational wavefunction $|\Psi_G\rangle = \hat{P}|\Psi_0\rangle$, where $|\Psi_0\rangle$ is the ground state of a free BCS-type variational Hamiltonian and $\hat{P}$ is a product of Gutzwiller and Jastrow factors that enforce the relevant local Hilbert space: one or two electrons per site for the t-J model, with doublon-holon processes for the Hubbard model. The argument runs by optimizing all hoppings, pairings, and Jastrow parameters with stochastic reconfiguration and comparing the optimized energies of the s±-wave ansatz (onsite, vertical-bond, and in-plane nearest-neighbor pairings) against the d_{$x^{2}$-$y^{2}$}-wave ansatz (in-plane nearest-neighbor pairings only). The pairing parameters are inspected after optimization, which is how the paper identifies the 3d_{$3z^{2}$-$r^{2}$} vertical bond as the leading pairing channel.

Extended reading notes

Core claim

On the paper's own terms, the discovery is an energy ordering established by unbiased variational optimization: in the two-orbital bilayer model with parameters from band-structure studies, the Jastrow-projected BCS ansatz with s±-wave pairing lowers the ground-state energy more than the analogous d_{$x^{2}$-$y^{2}$}-wave ansatz. For the Hubbard model, s± is favored for Hund's coupling above about 0.2 eV; for the extended t-J model, it wins over the entire parameter range studied. The optimized pairing amplitudes show the dominant component is $\Delta^z_{ij}$ on the vertical bond between 3d_{$3z^{2}$-$r^{2}$} orbitals, with onsite pairings of opposite sign on the two orbitals, consistent with pair-hopping. The result bridges the weak-to-moderate coupling regime, where renormalization-group studies find s±, and the strong-coupling regime, where the t-J model applies. A secondary finding is that intra-orbital double occupancy stays finite as U grows, so the electrons do not become well-defined local moments despite strong correlations.

Load-bearing premise

Everything rests on the variational wavefunction being flexible enough: the calculation only lets pairs form on a fixed set of bonds (onsite, vertical, and in-plane nearest-neighbor), so if a pairing channel outside that set would lower the energy, the s± versus d-wave comparison is incomplete.

Editorial extensions

If this is right

If s± is the ground state, the superconducting gap must change sign between the α pocket around Γ and the β/γ pockets around M, a signature phase-sensitive experiments can look for.
The dominance of the vertical 3d_{3z^2-r^2} pairing means interlayer coupling is not a minor correction: materials or pressures that alter the Ni-O-Ni vertical bonding should strongly affect the critical temperature.
The consistency between weak-coupling renormalization-group and strong-coupling variational results argues that s±-wave pairing survives across the correlation regime, not just at weak or strong coupling.
Finite double occupancy even at large U implies the carriers retain itinerant character, so treatments that start from a local-moment picture will miss part of the pairing physics.

Reading between the lines

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

The variational ansatz restricts pairing to a fixed bond set; adding d_{xy} or longer-range extended-s channels could lower the energy further, in which case the paper's s± versus d-wave comparison would be incomplete rather than wrong.
The paper's 'not too weak Hund's coupling' threshold (about 0.2 eV in the Hubbard model) makes the pairing-symmetry question partly an experimental one: a reliable determination of the Hund's coupling in La3Ni2O7 would say which side of the competition the real material sits on.
Because the leading pair lives on the vertical 3d_{3z^2-r^2} bond, tuning interlayer hybridization through chemical pressure or rare-earth substitution in the R3Ni2O7 family gives a testable route to confirm the mechanism.
The finite double occupancy at strong coupling suggests spin-fluctuation theories built purely on local moments may underestimate pairing strength; an orbital-resolved susceptibility measurement could test the itinerant picture.

Original abstract

We investigate the pairing symmetry in the novel superconductor La$_3$Ni$_2$O$_7$ under pressure by the non-perturbative variational quantum Monte Carlo. Within the bilayer Hubbard model and extended $t-J$ model with two orbitals in the $E_g$ doublet, we find the local strong correlation triggers $s_\pm$-wave Cooper pairing, with sign change of the gap function among the various Fermi pockets, while the $d_{x^2-y^2}$-wave pairing is generically disfavored. This is in agreement with the results from functional renormalization group applied in the weak up to moderate correlation limit. We find the 3d$_{3z^2-r^2}$ orbital plays a leading role in the superconducting pairing. We also demonstrate the finite intra-orbital double occupancy even in the strong correlation limit, shedding light on the itinerant versus local moment picture of the electrons in this material.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,455 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.97 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We investigate the pairing symmetry in the novel superconductor La$_3$Ni$_2$O$_7$ under pressure by the non-perturbative variational quantum Monte Carlo. Within the bilayer Hubbard model and extended $t-J$ model with two orbitals in the $E_g$ doublet, we find the local strong correlation triggers $s_\pm$-wave Cooper pairing, with sign change of the gap function among the various Fermi pockets, while the $d_{x^2-y^2}$-wave pairing is generically disfavored. This is in agreement with the results from functional renormalization group applied in the weak up to moderate correlation limit. We find the 3d$_{3z^2-r^2}$ orbital plays a leading role in the superconducting pairing. We also demonstrate the finite intra-orbital double occupancy even in the strong correlation limit, shedding light on the itinerant versus local moment picture of the electrons in this material.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We investigate the pairing symmetry in the novel superconductor La$_3$Ni$_2$O$_7$ under pressure by the non-perturbative variational quantum Monte Carlo.	contribution 0.98	0.95	0.06	asserted 0.88
s1	Within the bilayer Hubbard model and extended $t-J$ model with two orbitals in the $E_g$ doublet, we find the local strong correlation triggers $s_\pm$-wave Cooper pairing, with sign change of the gap function among the various Fermi pockets, while the $d_{x^2-y^2}$-wave pairing is generically disfavored.	result 1.00	0.97	0.83	asserted 1.00
s2	This is in agreement with the results from functional renormalization group applied in the weak up to moderate correlation limit.	interpretation 0.59	0.07	0.06	asserted 0.72
s3	We find the 3d$_{3z^2-r^2}$ orbital plays a leading role in the superconducting pairing.	result 1.00	0.96	0.19	asserted 1.00
s4	We also demonstrate the finite intra-orbital double occupancy even in the strong correlation limit, shedding light on the itinerant versus local moment picture of the electrons in this material.	result 0.81	0.96	0.06	asserted 1.00

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** (a) Energy band dispersions along the high-symmet

**Figure 2.** (a) Schematic plot of the pairing potentials (or va

**Figure 3.** VQMC results of the t − J model. The energies of the s±-wave and dx 2−y 2 -wave pairings relative to the normal state are plotted versus U by fixing JH = 0.5 eV in (a), and versus JH by fixing U = 7 eV in (b). The corresponding optimized variational pairing parameters for the s±-wave are plotted versus U in (c) and versus JH in (d), respectively. the projection operator for n-electrons on orbital a and site i, Q0e,ia = (1−nia↑)(1−nia↓), Q2e,ia = nia↑nia↓, Q1e,ia = nia↑(1−nia↓) + (1−nia↑)nia↓, (4) and Qne,i is the projection operator for total charge n-electrons on site i, Qne,i = n…

**Figure 4.** VQMC results of the Hubbard model. The energies of t

**Figure 5.** Finite size effect with lattice size L = 6 to 12 for the energies of the s±-wave and dx 2−y 2 -wave pairings at U = 7 eV and JH = 0.5 eV

**Figure 6.** The optimized variational hopping and onsite ener

**Figure 7.** The optimized Jastrow factors for the Hubbard mode

Figures reproduced from arXiv:2505.07341 by the authors.

References

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However, we held a different viewpoint that the interlayer pairing $s$-wave SC is induced by the $3d_{x^2-y^2}$ orbital, driven by the strong interlayer superexchange interaction.

Interplay of two $E_g$ orbitals in Superconducting La$_3$Ni$_2$O$_7$ Under Pressure · 0.91

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A large number of experimental [ref] and theoretical [ref] studies have been conducted to investigate the superconductivity, density wave and normal state electronic properties.

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The energy gains of the s± and dx2−y2 pairings relative to the normal state are plotted versus U (ﬁxing JH = 0.5 eV) in Fig. 4(a) and versus JH (ﬁxing U = 7 eV) in Fig. 4(b). We ﬁnd the s±-wave pair- ing always saves more energy than the dx2−y2 -wave pairing as long as JH > 0.2 eV (very likely in La 3Ni2O7). This is one of the main results of the present ...

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