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arxiv: 2606.08181 · v1 · pith:C55WKEFDnew · submitted 2026-06-06 · ❄️ cond-mat.supr-con · cond-mat.str-el

Microscopic mechanism of high-temperature superconductivity revealed by ab initio studies on hole-doped multilayer cuprates HgBa₂Ca₂Cu₃O₈ under pressure

Ryui Kaneko , Masatoshi Imada This is my paper

Pith reviewed 2026-06-27 19:06 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.str-el
keywords high-temperature superconductivitycupratesab initio Hamiltoniand-wave pairingMott insulatoremergent attractionHg1223pressure dependence
0
0 comments X

The pith

Strong local repulsion U generates an emergent local attraction that drives d-wave superconductivity in Hg1223.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper solves ab initio Hamiltonians for the triple-layer cuprate HgBa2Ca2Cu3O8 using a variational neural-network solver. It reproduces the dome-like pressure dependence of the d-wave order parameter and estimated Tc in agreement with experiment. The central finding is that the originally strong on-site repulsion U produces an effective instantaneous attraction once carriers are doped, because doping releases fluctuating doubly occupied sites that were trapped in the Mott-insulator false vacuum. This attraction coexists with antiferromagnetic order in the multilayer geometry and strengthens further under pressure through reduced intersite repulsion V.

Core claim

The pairing mechanism is identified as the emergent local attraction counterintuitively generated from the originally strong local repulsion U. The emergent attraction is interpreted from attraction from reduced repulsion, originating from the release of the fluctuating doubly-occupied sites characterized from the false vacuum in the Mott insulator to the double-occupation-free d-wave SC states upon carrier doping. This instantaneous attraction is in contrast with the conventional BCS SC mediated by bosonic glues.

What carries the argument

Emergent local attraction generated from strong on-site repulsion U by the release of doubly occupied sites from the Mott false vacuum into d-wave states.

Load-bearing premise

The variational neural-network solver accurately captures the ground-state properties of the ab initio Hamiltonian for the multi-layer system, including the pressure dependence of the d-wave order parameter, without significant bias from the chosen ansatz.

What would settle it

A direct experimental probe that shows whether the energy cost of creating double occupancy is lower in the superconducting state than in the normal state at the same doping.

Figures

Figures reproduced from arXiv: 2606.08181 by Masatoshi Imada, Ryui Kaneko.

Figure 1
Figure 1. Figure 1: FIG. 1. Crystal structure of Hg1223 and directions of primitive [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Inner- and outer-layer hole densities as a function of total [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Doping concentration dependence of the SC order parameter [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Size dependent SC order parameters [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Size extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Inner- and outer-layer hole densities as a function of total [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Pressure dependence of the SC order parameter ( [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Comparison of pressure dependence of [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Materials dependence of SC critical temperature [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Coulomb interaction dependence of the SC order parameter [PITH_FULL_IMAGE:figures/full_fig_p010_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Product of the attraction [PITH_FULL_IMAGE:figures/full_fig_p011_15.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Doping concentration ( [PITH_FULL_IMAGE:figures/full_fig_p012_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Doping concentration dependence of the SC correlation [PITH_FULL_IMAGE:figures/full_fig_p017_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Doping concentration dependence of the SC correlation [PITH_FULL_IMAGE:figures/full_fig_p017_19.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. Size extrapolation of the SC correlation averaged over long [PITH_FULL_IMAGE:figures/full_fig_p017_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23. Size extrapolation of the SC correlation averaged over long [PITH_FULL_IMAGE:figures/full_fig_p018_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24. Variance extrapolation of energies of ground-state candidate [PITH_FULL_IMAGE:figures/full_fig_p018_24.png] view at source ↗
Figure 26
Figure 26. Figure 26: FIG. 26. Onsite interaction ( [PITH_FULL_IMAGE:figures/full_fig_p019_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: FIG. 27. Onsite interaction, ( [PITH_FULL_IMAGE:figures/full_fig_p020_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: FIG. 28. Doping concentration ( [PITH_FULL_IMAGE:figures/full_fig_p021_28.png] view at source ↗
read the original abstract

Triple-layer cuprate superconductor $\mathrm{HgBa_2Ca_2Cu_3O_8}$ (Hg1223) keeps the record of the highest superconducting (SC) critical temperature $T_{c}\sim 134$K among all the existing materials at ambient pressure. $T_{c}$ further increases under pressure up to $T_{c}\sim 160$K. However, its microscopic mechanism remains to be elucidated. We solve {\it ab initio} Hamiltonians for Hg1223 using a variational solver supplemented by a neural network. The pressure dependence of the $d$-wave SC order parameter and estimated $T_{c}$ show a dome-like structure in essential agreement with the experimental indications. The origin of the strong SC amplitude at ambient pressure is identified as strong local Coulomb repulsion $U$ attributed to poor screening. Further increase in $T_{c}$ under pressure is understood from interplay of three elements, namely increased electron hopping $t$, decreased $U$ and more importantly, strongly reduced offsite Coulomb repulsion $V$ with increasing pressure. Pairing mechanism is identified as the emergent local attraction counterintuitively generated from the originally strong local repulsion $U$. The emergent attraction is interpreted from ``attraction from reduced repulsion'', originating from the release of the fluctuating doubly-occupied sites characterized from the ``false vacuum'' in the Mott insulator to the double-occupation-free $d$-wave SC states upon carrier doping. This instantaneous attraction is in contrast with the conventional BCS SC mediated by bosonic glues. The local attraction is consistent with the electron fractionalization supported in experimental analyses. The coexistence of the SC and antiferromagnetic order is also demonstrated as a characteristic feature of the multi-layer system. The microscopic understanding of Hg1223 offers a new route explicitly using this emergent attraction to design and optimize SC materials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript solves ab initio Hamiltonians for the triple-layer cuprate HgBa2Ca2Cu3O8 (Hg1223) with a variational neural-network solver. It reports that the pressure dependence of the d-wave superconducting order parameter and estimated Tc exhibit a dome-like structure in essential agreement with experiment. The origin of strong SC amplitude is attributed to strong local Coulomb repulsion U due to poor screening, with further Tc increase under pressure arising from increased t, decreased U, and strongly reduced V. The pairing mechanism is identified as emergent local attraction generated from originally strong U via release of fluctuating doubly-occupied sites from the Mott 'false vacuum' into double-occupation-free d-wave SC states upon doping; this is contrasted with conventional BCS mediation by bosonic glues. Coexistence of SC and antiferromagnetic order is also reported as a multi-layer feature.

Significance. If the variational results faithfully represent the ground state, the work would offer a concrete microscopic route to high-Tc cuprate superconductivity based on instantaneous emergent attraction rather than retarded bosonic exchange, with direct implications for material optimization under pressure. The explicit use of ab initio parameters and the reported agreement on pressure dependence of the order parameter would strengthen the case for this mechanism over phenomenological models.

major comments (3)
  1. [Methods / abstract claim on variational solver] The central claim that the variational neural-network solver accurately captures the ground-state d-wave order parameter, double occupancy, and their pressure dependence in the multi-layer ab initio Hamiltonian (and thereby identifies the 'attraction from reduced repulsion' mechanism) lacks supporting validation. No convergence checks, error bars on the order parameter, or benchmarks against unbiased methods (e.g., DMRG on smaller clusters) are supplied to rule out ansatz bias in the multi-layer setting with interlayer couplings.
  2. [Results on pressure dependence and Tc estimation] The estimated Tc is obtained from the computed d-wave order parameter whose pressure dependence is stated to match experiment. It is unclear from the presented results whether any Hamiltonian parameters or neural-network training procedure incorporate information from the same experimental Tc data, which would introduce circularity into the reported agreement.
  3. [Discussion of pairing mechanism] The interpretation of emergent local attraction as arising specifically from release of doubly-occupied sites from the Mott false vacuum relies on the wavefunction representation of fluctuations; without explicit checks that this feature is robust to changes in the variational ansatz or truncation, it remains possible that the interpretation is an artifact of the chosen solver rather than a property of the Hamiltonian.
minor comments (2)
  1. [Methods] Notation for the ab initio parameters (t, U, V) and their pressure dependence should be defined explicitly with numerical values or functional forms in a dedicated table or section for reproducibility.
  2. [Results] The abstract states 'essential agreement' with experiment but supplies no quantitative measure (e.g., RMS deviation or overlap of dome shapes); this should be added to the results section.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below, providing clarifications and indicating where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: The central claim that the variational neural-network solver accurately captures the ground-state d-wave order parameter, double occupancy, and their pressure dependence in the multi-layer ab initio Hamiltonian (and thereby identifies the 'attraction from reduced repulsion' mechanism) lacks supporting validation. No convergence checks, error bars on the order parameter, or benchmarks against unbiased methods (e.g., DMRG on smaller clusters) are supplied to rule out ansatz bias in the multi-layer setting with interlayer couplings.

    Authors: We agree that additional validation details would improve the presentation. In the revised manuscript we will add convergence tests with respect to neural-network depth/width and Monte Carlo sampling statistics, together with error bars on the order parameter obtained from independent optimization runs. Direct DMRG benchmarks on the full multi-layer Hamiltonian with interlayer couplings are computationally prohibitive at the relevant system sizes; however, we will include comparisons on smaller single-layer clusters where DMRG is feasible and reference prior benchmarks of the same variational ansatz on related Hubbard models. These additions address the concern without altering the central conclusions. revision: partial

  2. Referee: The estimated Tc is obtained from the computed d-wave order parameter whose pressure dependence is stated to match experiment. It is unclear from the presented results whether any Hamiltonian parameters or neural-network training procedure incorporate information from the same experimental Tc data, which would introduce circularity into the reported agreement.

    Authors: All Hamiltonian parameters (hopping t, onsite U, offsite V, etc.) are taken directly from ab initio downfolding calculations with no adjustment to experimental Tc values. The variational neural-network optimization minimizes the energy of this fixed ab initio Hamiltonian; no experimental data enter the training. The Tc estimate is obtained from the computed order parameter via a standard mean-field scaling relation that likewise contains no experimental input. The reported agreement with the experimental pressure dependence is therefore a post-diction. We will add an explicit statement clarifying this point in the revised manuscript. revision: yes

  3. Referee: The interpretation of emergent local attraction as arising specifically from release of doubly-occupied sites from the Mott false vacuum relies on the wavefunction representation of fluctuations; without explicit checks that this feature is robust to changes in the variational ansatz or truncation, it remains possible that the interpretation is an artifact of the chosen solver rather than a property of the Hamiltonian.

    Authors: The physical picture is extracted from the optimized wave function’s double-occupancy correlations and their evolution with doping and pressure. We have verified that the qualitative trend in double occupancy persists across several neural-network architectures (different depths and activation functions). A exhaustive scan of entirely different ansatz families lies beyond the scope of the present work, but the same emergent-attraction mechanism appears in simpler variational treatments of the Hubbard model, suggesting it is tied to the Hamiltonian rather than the specific solver. We will add a short paragraph discussing ansatz sensitivity in the revised discussion section. revision: partial

Circularity Check

0 steps flagged

No circularity: ab initio Hamiltonian solved variationally, results compared to experiment as validation

full rationale

The paper solves ab initio-derived Hamiltonians for Hg1223 using a variational neural-network solver. The d-wave order parameter and estimated Tc are computed outputs whose pressure dependence is then compared to experimental indications. No equations or text in the provided abstract reduce the central claims (emergent attraction from reduced repulsion, false-vacuum release) to fitted inputs or self-citations by construction. The derivation chain remains independent of the target observables.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; the central claim rests on the unstated accuracy of the ab initio downfolding to an effective Hamiltonian and on the variational neural-network solver faithfully representing the ground state of that Hamiltonian. No explicit free parameters, axioms, or invented entities can be extracted from the abstract alone.

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