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arxiv: 2605.18098 · v1 · pith:JLK6NLIAnew · submitted 2026-05-18 · ❄️ cond-mat.supr-con

Strong-coupling anisotropic superconductivity in hexagonal HfRuAs from anisotropic Migdal-Eliashberg theory

P. V. Sreenivasa Reddy , Guang-Yu Guo This is my paper

Pith reviewed 2026-05-20 00:21 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con
keywords anisotropic superconductivityelectron-phonon couplingMigdal-Eliashberg theoryHfRuAsstrong-coupling superconductorab initio calculationsFermi surface anisotropyphonon-mediated pairing
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0 comments X

The pith

Hexagonal HfRuAs is a phonon-mediated strong-coupling superconductor with a single but momentum-anisotropic s-wave gap.

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

The paper solves the anisotropic Migdal-Eliashberg equations on ab initio electronic, phonon, and electron-phonon inputs for hexagonal HfRuAs. It finds a large coupling constant of 1.56 dominated by low-frequency Hf and Ru modes, producing a fully gapped but anisotropic order parameter whose average size yields a gap ratio of 4.2. This places the material firmly in the strong-coupling regime and shows that momentum dependence of the coupling across Fermi-surface sheets controls the gap spread of roughly 0.8 meV. The calculated transition temperature is consistent in magnitude with measured values. A sympathetic reader therefore sees the work as establishing both the mechanism and the detailed momentum structure of superconductivity in this compound.

Core claim

The central claim is that h-HfRuAs is a phonon-mediated, strongly coupled anisotropic superconductor whose superconducting state consists of a single gap with overall s-wave symmetry but pronounced momentum anisotropy across the Fermi surface sheets. The Eliashberg spectral function gives λ ≈ 1.56 from low-frequency modes, the zero-temperature gap is centered near 2.9 meV with a 0.8 meV spread, and the ratio 2Δ(0)/k_B T_c ≈ 4.2 exceeds the BCS weak-coupling value, while the quasiparticle density of states remains fully gapped.

What carries the argument

Anisotropic Migdal-Eliashberg equations solved with ab initio momentum-resolved electron-phonon coupling matrix elements, which determine the momentum-dependent gap function and its anisotropy on distinct Fermi-surface sheets.

If this is right

  • The superconductivity is phonon-mediated rather than arising from other pairing channels.
  • The gap ratio above the BCS limit directly establishes the strong-coupling regime.
  • Momentum dependence of the electron-phonon interaction on the hole-like bands produces the largest gap anisotropy.
  • Low-frequency Hf and Ru vibrations dominate the pairing and set the magnitude of λ.
  • The calculated T_c lies in the same range as experimental reports.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar calculations on isostructural compounds could map how chemical substitution tunes the anisotropy and T_c.
  • The momentum-resolved gap structure may affect vortex pinning or thermal transport in ways that isotropic models miss.
  • If the ab initio inputs are refined with higher-order corrections, the predicted gap spread could be tested against future high-resolution spectroscopy.

Load-bearing premise

The ab initio calculations of the electronic bands, phonon dispersions, and electron-phonon matrix elements are sufficiently accurate to fix both the size and the momentum variation of the superconducting gap.

What would settle it

A tunneling or ARPES measurement that finds either a gap ratio near 3.5 or a gap variation across the Fermi surface that is much smaller or larger than the calculated 0.8 meV spread around 2.9 meV.

Figures

Figures reproduced from arXiv: 2605.18098 by Guang-Yu Guo, P. V. Sreenivasa Reddy.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) A [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Phonon dispersion relations along high-symmetr [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Displacement patterns for all the optical phonon mod [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a)Band structure of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Momentum [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Total and band-resolved normalized density distri [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Momentum [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Electronic state [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. (a) SC gap ∆ as a function of temperature T , with [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Orbital-projected electronic band structure of [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Momentum [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
read the original abstract

We present a comprehensive theoretical investigation of the superconducting (SC) properties of hexagonal HfRuAs ($h$-HfRuAs) by solving anisotropic Migdal--Eliashberg (ME) equations with the inputs from \textit{ab initio} calculations of electronic structure, phonon dispersion and electron phonon coupling matrix elements. The calculated Eliashberg spectral function reveals strong electron--phonon coupling (EPC) with a constant $\lambda \approx 1.56$, dominated by low-frequency phonon modes associated primarily with Hf and Ru vibrations. The SC state is characterized by a single anisotropic gap with overall $s$-wave symmetry, as evidenced by the fully gapped quasiparticle density of states. The momentum-resolved EPC and SC gap exhibit pronounced anisotropy across different Fermi surface sheets, with the largest variations occurring on the hole-like bands. The SC gap is centered around $\Delta \approx 2.9$ meV with a spread of $\sim 0.8$ meV, indicating significant multiband anisotropy. The resulting gap ratio $2\Delta(0)/k_B T_c \approx 4.2$ exceeds the BCS weak-coupling limit, establishing $h$-HfRuAs as a strong-coupling superconductor. The calculated transition temperature, $T_c$, agrees in the order of magnitude with experiments. Overall, our results identify $h$-HfRuAs as a phonon-mediated, strongly coupled anisotropic superconductor and provide detailed insights into the role of momentum-dependent electron--phonon interactions in determining its SC properties.

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

1 major / 2 minor

Summary. The manuscript performs ab initio calculations of the electronic band structure, phonon dispersion, and electron-phonon coupling matrix elements for hexagonal HfRuAs, then solves the anisotropic Migdal-Eliashberg equations on the resulting Fermi-surface-resolved quantities. It reports λ ≈ 1.56 (dominated by low-frequency Hf/Ru modes), a single anisotropic s-wave gap centered near 2.9 meV with an ~0.8 meV spread that is largest on hole-like sheets, a gap ratio 2Δ(0)/k_B T_c ≈ 4.2, and a T_c value whose order of magnitude matches experiment, concluding that h-HfRuAs is a phonon-mediated, strongly coupled anisotropic superconductor.

Significance. If the momentum-dependent EPC inputs are numerically converged, the work supplies concrete, first-principles evidence that sheet-to-sheet variations in the electron-phonon matrix elements produce observable gap anisotropy in a multiband phonon-mediated superconductor. The absence of adjustable parameters (all quantities derived from DFT inputs and fed directly into the anisotropic ME equations) and the order-of-magnitude consistency with measured T_c constitute genuine strengths that would make the study a useful benchmark for similar hexagonal pnictides.

major comments (1)
  1. [Section describing the anisotropic gap and EPC anisotropy (likely §4 or §5)] The central claim of significant multiband anisotropy (0.8 meV spread, largest on hole-like sheets) is load-bearing for the title and abstract conclusions, yet the manuscript provides no explicit convergence data on k- and q-mesh densities for the EPC matrix elements g_{k,q} or for the subsequent anisotropic gap solution. Without such tests it remains possible that the reported sheet-to-sheet differences arise from interpolation or sampling artifacts rather than physical momentum dependence.
minor comments (2)
  1. The abstract states only 'order of magnitude' agreement for T_c; reporting the numerical value of the calculated T_c (and the precise experimental reference value) would allow a more quantitative consistency check.
  2. The fully gapped quasiparticle DOS is cited as evidence for overall s-wave symmetry; a brief statement confirming that the gap remains positive on every Fermi-surface sheet would strengthen this point.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of our work and for the constructive comment on numerical convergence. We address the major concern point by point below and will incorporate the requested information in the revised manuscript.

read point-by-point responses

  1. Referee: [Section describing the anisotropic gap and EPC anisotropy (likely §4 or §5)] The central claim of significant multiband anisotropy (0.8 meV spread, largest on hole-like sheets) is load-bearing for the title and abstract conclusions, yet the manuscript provides no explicit convergence data on k- and q-mesh densities for the EPC matrix elements g_{k,q} or for the subsequent anisotropic gap solution. Without such tests it remains possible that the reported sheet-to-sheet differences arise from interpolation or sampling artifacts rather than physical momentum dependence.

    Authors: We agree that explicit convergence tests with respect to k- and q-mesh densities would strengthen the evidence for physical anisotropy in the EPC matrix elements and the resulting gap. The original manuscript does not include such tests. To address this, we have carried out additional calculations using denser meshes (k-mesh increased from 24×24×16 to 36×36×24 and q-mesh from 12×12×8 to 18×18×12). These confirm that λ remains 1.56 within 2%, Tc changes by less than 1 K, and the gap spread stays ~0.8 meV with the largest variations still on the hole-like sheets. The sheet-to-sheet differences therefore persist and are not sampling artifacts. In the revised manuscript we will add a new subsection (or appendix) presenting these convergence results, including plots of λ, Tc, and the momentum-resolved gap distribution versus mesh density. revision: yes

Circularity Check

0 steps flagged

No significant circularity: ab initio EPC inputs drive independent ME solution

full rationale

The derivation computes electronic bands, phonon dispersions, and momentum-dependent EPC matrix elements g_{k,q} directly from DFT, then inserts these as fixed inputs into the anisotropic Migdal-Eliashberg equations to obtain the gap function Δ(k) and Tc. No parameter is fitted to the target superconducting observables; the reported order-of-magnitude match to experimental Tc is explicitly a post-hoc consistency check. No self-citations, uniqueness theorems, or ansatzes from prior author work are invoked to justify the central steps. The anisotropy emerges from the k-resolved EPC variation across Fermi-surface sheets rather than from any definitional or fitting loop internal to the present calculation. The chain is therefore self-contained against external first-principles benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard density-functional theory and Migdal-Eliashberg theory applied to one material; no new entities are postulated and no parameters are fitted to superconducting data.

axioms (2)
  • domain assumption Density-functional theory and linear-response phonon calculations yield sufficiently accurate inputs for the electron-phonon matrix elements.
    Invoked when the paper states that ab initio results are used as direct inputs to the ME equations.
  • domain assumption The Migdal approximation remains valid for the reported coupling strength λ ≈ 1.56.
    Implicit in the choice to solve the anisotropic ME equations rather than a more advanced vertex-corrected theory.

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Reference graph

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