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REVIEW 3 major objections 4 minor 82 references

Multiphase superconductivity in PdBi2

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read In-plane fields switch β-PdBi2 from s-wave to nodal superconductivity at about 0.2 T.

desk verdict A solid experimental case for a field-induced superconducting transition in beta-PdBi2, with the nodal p-wave assignment overreaching the data. read the letter →

arxiv 2411.09239 v1 pith:VA2DKIVH submitted 2024-11-14 cond-mat.supr-con

Lewis Powell , Wenjun Kuang , Gabriel Hawkins-Pottier , Rashid Jalil , John Birkbeck , Ziyi Jiang , Minsoo Kim , Yichao Zou
show 8 more authors
Sofiia Komrakova Sarah Haigh Ivan Timokhin Geetha Balakrishnan Andre K. Geim Niels Walet Alessandro Principi Irina V. Grigorieva
This is my paper · ORCID
classification cond-mat.supr-con
keywords unconventionalsuperconductivityfield-inducedphasetransitionnodalpairingp-wavespin-orbitcouplingtunnellingspectroscopybeta-PdBi2hiddenspinpolarization
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

The paper reports that β-PdBi2, a layered non-magnetic superconductor with strong spin-orbit coupling, switches pairing symmetry inside the superconducting state when an in-plane magnetic field is applied. Below a transition field of roughly 0.2 T, tunnelling spectra match a conventional s-wave superconductor described by BCS theory with pair breaking; above it, the spectra become V-shaped with rapidly rising zero-bias conductance, the signature of a nodal gap that the authors fit with a p-wave order parameter $\Delta\cos\theta$. The proposed mechanism is that locally broken inversion symmetry locks electron spins to momentum, so an in-plane field anisotropically splits the spin-locked Fermi surfaces and makes equal-spin p-wave pairing energetically favourable. If correct, the result is a magnetic-field-driven transition between two superconducting phases in a non-magnetic material, and it reconciles theoretical predictions of unconventional multigap superconductivity in β-PdBi2 with earlier experiments that detected only a single s-wave gap.

What carries the argument

The argument turns on two fitting models for the tunnelling conductance and one microscopic model for the transition. Below $B^*$ the spectra are described by the standard Dynes-broadened BCS density of states within the Maki theory of a thin s-wave film in a parallel field, whose only free parameters are the gap $\Delta$ and the pair-breaking strength $\zeta$; the extracted $\zeta(B)$ matches the theory only if an apparent s-wave critical field of about 0.25 T is used. Above $B^*$ the spectra are described by the nodal density of states $N_S/N_N = \mathrm{Re}\big[(E+i\Gamma)\Delta^{-1}\arcsin(\Delta/(E+i\Gamma))\big]$ for a p-wave order parameter $\Delta\cos\theta$, with $\Gamma$ absorbing pair breaking. The microscopic model is a minimal bilayer Rashba Hamiltonian on the Bi sublattices, in which globally centrosymmetric layers are locally non-centrosymmetric; two competing interactions, intra-sublayer $U$ (s-wave) and inter-sublayer $V$ (spin-triplet), enter a mean-field free energy $F(\psi,\eta,B,T)$ whose minimisation yields a first-order transition from the s-wave order parameter $\psi$ to the spin-polarised p-wave order parameter $\eta$ at a temperature-independent field $B^*$, with the triplet gap nodes aligned along the field.

What would settle it

Fit the spectra above $B^*$ with a nodeless anisotropic s-wave gap, using the same two-parameter freedom as the p-wave fit, and compare the residuals; if that model reproduces the V-shaped conductance as well as the nodal density of states does, the nodal-pairing conclusion is not established. A bulk thermodynamic probe such as specific heat or superfluid density in a parallel field should also show a distinct feature at $B^*$ if the transition is first-order, and its absence would point to a continuous crossover.

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Extended reading notes

Core claim

β-PdBi2 is a single-gap s-wave superconductor at zero field, but tunnelling spectroscopy on thin crystals in planar superconductor-insulator-normal metal junctions reveals a sharp, field-driven change inside the superconducting state. For in-plane fields below $B^*\approx 0.2$ T the conductance spectra are quantitatively described by the standard theory of a thin s-wave film in a parallel field, with an extracted gap $\Delta(B)$ that collapses towards an apparent critical field of about 0.25 T. At $B^*$ the spectra abruptly become V-shaped, the zero-bias conductance rises almost linearly instead of staying at zero as it would until roughly 60-70% of $B_{c2}$ for a conventional film, and the quasiparticle peaks persist; the authors show that this behaviour is inconsistent with any realistic s-wave pair-breaking strength but is accurately fit by the density of states of a nodal p-wave gap. A kink in the measured in-plane upper critical field $B_{c2}^\parallel(T)$ marks the same boundary, and no transition appears for out-of-plane fields. The authors attribute the effect to hidden spin-momentum locking: an in-plane Zeeman field splits the Rashba-split bands anisotropically, and a minimal mean-field model with competing s-wave and spin-polarised triplet channels then shows a first-order transition to p-wave pairing at a temperature-independent $B^*$, with the two phases coexisting over a narrow field window.

Load-bearing premise

The high-field state is identified as nodal because the V-shaped spectra above $B^*$ are fit with a p-wave density of states, but the paper does not test whether an anisotropic but nodeless s-wave gap, which can also produce V-shaped conductance, fits the same data equally well.

Editorial extensions

If this is right

  • If the transition is first-order, the s-wave and p-wave phases should coexist over a narrow field range near $B^*$; the authors note that spectra in this window are fit equally well by both models, consistent with coexistence of the two order parameters.
  • The transition field $B^*$ is roughly temperature-independent, and it moves to lower fields in thinner crystals alongside a suppressed $T_c$ and enhanced in-plane $B_{c2}$, a trend the authors connect to surface states favouring p-wave pairing near the boundaries.
  • The s-wave picture fails quantitatively above $B^*$: reproducing the spectra within the Maki theory demands pair-breaking strengths far beyond the values the same theory predicts, whereas the nodal p-wave form fits with only $\Delta$ and $\Gamma$ as parameters.
  • Out-of-plane fields produce no transition and leave s-wave pairing intact, so the effect is tied to the spin texture of the bands rather than to ordinary orbital pair breaking, and the same model explains why triplet pairing is unfavourable for out-of-plane fields.
  • A finite-momentum FFLO state is ruled out as an alternative explanation because both $B^*$ and $B_{c2}$ lie well below the Pauli paramagnetic limit for this material.

Reading between the lines

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

  • The model predicts the p-wave nodes to align with the in-plane field direction, so rotating the field within the plane and tracking the tunnelling spectra would test whether the node orientation follows the field — a signature the paper does not examine.
  • The claimed coexistence of the two phases between roughly $B^*$ and $2B^*$ implies spatial inhomogeneity; local probes such as scanning tunnelling microscopy across that field window could image normal or nodal domains and test the first-order picture directly.
  • The thickness dependence suggests surface states participate: if hybridisation with topological surface states favours p-wave pairing, $B^*$ should continue to decrease for thinner crystals and eventually vanish at the two-dimensional limit.
  • A bulk thermodynamic measurement, such as specific heat in a parallel field, would locate the transition independently of the surface-sensitive tunnelling signal and could distinguish a sharp first-order jump from a continuous crossover.
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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 paper reports tunnelling spectroscopy and transport measurements on thin exfoliated crystals of the layered superconductor β-PdBi2, using SIN tunnel junctions with hBN barriers and few-layer graphene electrodes. The central observation is a sharp, reproducible change in the tunnelling spectra as an in-plane magnetic field is increased above B* ≈ 0.2 T: the spectra evolve from a fully gapped s-wave form described by Maki theory to 'V'-shaped spectra with rapidly increasing zero-bias conductance, the extracted gap parameter shows a pronounced kink and re-entrant increase, and the in-plane upper critical field Bc2(T) exhibits a kink. The authors interpret these findings as a field-induced first-order transition from conventional s-wave pairing to a nodal (most likely p-wave triplet) superconducting state, and support this with a minimal tight-binding model incorporating hidden Rashba spin–orbit coupling, in which an in-plane field stabilizes spin-polarized triplet pairing. They also report thickness-dependent T_c and B* and discuss the relation to topological surface states and to other multiphase superconductors.

Significance. If the nodal-pairing assignment is correct, the result is significant: it would place β-PdBi2 among the very few materials displaying a magnetic-field-driven transition between superconducting phases of different pairing symmetry, in a non-magnetic, strongly spin-orbit-coupled system, and it would reconcile earlier theoretical predictions of multigap or unconventional pairing with the single s-wave gap seen in most prior experiments. The paper has notable strengths: the tunnelling data are of high quality, the behaviour is reproduced across five devices, the comparison to Maki's s-wave theory is careful and quantitative, and the authors provide a testable theoretical framework as well as their fitting code as supplementary material. However, the strength of the central claim is currently limited by the underdetermination of the high-field order parameter symmetry from the tunnelling fits, and by the circular element in the theory calibration, as detailed in the major comments.

major comments (3)
  1. [Results, Fig. 3b and Supplementary Fig. 5b] The identification of the high-field phase as nodal (p-wave) rests on fitting the V-shaped spectra with the line-node DoS of Eq. (3), while the only alternative considered is the Maki s-wave model. A nodeless anisotropic s-wave gap, for example Δ(θ)=Δ0(1+r cos 2θ) with Δ_min>0, convolved with the same thermal factor and lifetime broadening Γ, would also produce V-shaped conductance with finite zero-bias conductance if Γ is comparable to or larger than Δ_min. This model has not been fitted to the data in Fig. 3b or Supplementary Fig. 5b, and the authors explicitly note that in the transitional region (B≈0.2–0.4 T) the s-wave Maki and nodal p-wave fits are equally good. The data therefore establish a sharp field-induced spectral change, but the assignment to nodal pairing is underdetermined by the fits shown. The Zeeman-field argument in the Discussion addresses thermodynamic competition between singlet pairing channels in the model, not the ability of an anisotropic singlet gap to mimic a V-shaped superconducting DoS in tunnelling.
  2. [Discussion, Eq. (6) and following] The theoretical 'prediction' of the transition field B* is not independent of the experiment: the interaction strengths U and V are fixed by inserting T_c^s≈3 K and T_c^p≈2.4 K into Eq. (6), where T_c^p is itself obtained as an extrapolation of the high-field branch of the experimentally measured Bc2(T) curve to B=0. The free-energy crossing in Fig. 4b then reproduces the B* that was already used as input. The model is therefore a consistency check of the two-phase interpretation rather than a parameter-free prediction. This should be stated explicitly, and the claim of a 'predicted first-order phase transition' should be softened accordingly.
  3. [Main text, Fig. 3a and Methods, 'Fitting tunnelling data'] The quantitative statement of a re-entrant increase in the order parameter above B* depends on the choice of the fitting model: above B*, Δ(B) is extracted from Eq. (3), a nodal-DoS expression, whereas below B* it is extracted from the Maki s-wave DoS. If the high-field state were instead a nodeless anisotropic s-wave state, the fit parameter in Eq. (3) would not correspond to the superconducting gap in the usual sense, and the reported discontinuity in Δ(B) would be an artifact of the model switch. The paper should present the raw spectral evolution alongside a comparative fit with an anisotropic s-wave gap, and the extracted Δ(B) should be clearly labelled as model-dependent.
minor comments (4)
  1. [Methods, 'Fitting tunnelling data'] The text says 'we numerically solved eqs. (8),(9)' but the equations in the main text are numbered (9) and (10); the cross-reference should be corrected.
  2. [Supplementary Note 2.1] The derivation of the line-node DoS in Eq. (S3) uses a 3D angular integration over dΩ_k, whereas the Fermi surface of the 2D-like band considered in the continuum model is a cylinder; the relation between the 3D angular average and the effective 2D line-node density of states should be clarified.
  3. [Fig. 2d] The two straight-line fits to B_c2^||(T) above and below the kink at ~0.5 T are presented visually without error bars or residuals; a quantitative measure of the quality of the two-line description (and, ideally, a statistical test against a single-line fit) would strengthen the claim of a kink.
  4. [Abstract and Discussion] The abstract states the transition is 'consistent with' nodal pairing, which is appropriately cautious; however, the Discussion uses stronger language ('the new phase takes over', 'fully p-wave') and the phrase 'the predicted first-order phase transition' in the penultimate paragraph overstates the status of the theory, which is calibrated to the same data and explicitly omits orbital depairing.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the field-induced transition is defined by direct tunnelling and transport data, and the theory is a calibrated consistency model whose B* is not fitted to the experimental B*; the main weakness is an untested anisotropic-s-wave alternative, which is underdetermination rather than circularity.

full rationale

The experimental claim of a field-induced transition rests on direct conductance measurements: the kink in Delta(B), the sharp increase in zero-bias conductance, the spectral shape change, and the kink in the in-plane Bc2(T) phase diagram are all extracted from data using standard Dynes and Maki fitting expressions. No equation in the paper defines the experimental transition field as an output of the fitted theory, so the central observation is not circular. The theoretical section estimates U and V from the measured Tc^s-wave = 3 K and the extrapolated Tc^p-wave = 2.4 K (text after eq. 6), then computes the free-energy crossing in Fig. 4. This makes the theoretical p-wave stability calculation a calibrated consistency check rather than a parameter-free prediction, but the experimental B* itself is not used as an input; the model's crossing field is expressed relative to the Pauli limit (~0.7 Bc2^s-wave) and is only compared qualitatively with the data, with orbital depairing added by assumption. Thus the theory does not reduce to the data by construction. The p-wave assignment is, however, underdetermined: the V-shaped spectra above B* are fit with the nodal p-wave DoS of eq. (3), while a nodeless anisotropic s-wave gap with lifetime broadening is never fit or excluded. That is an incomplete model-selection argument and a real correctness risk, but it is not circularity, because the p-wave fit is not derived from the same parameters used to infer the transition. The only overlapping self-citation (ref. 10) appears in passing and in a speculative surface-state explanation; it is not load-bearing for the central claim. Score 2 reflects the minor non-load-bearing self-citation and the self-consistency nature of the theory, not demonstrated circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central experimental claim rests on standard tunnelling and BCS analysis. The theoretical explanation adds model parameters (U, V) estimated from the same experimental data, and simplifying assumptions (neglect of orbital depairing, circular Fermi surface, two-channel interaction) that are not independently benchmarked.

free parameters (4)
  • UN_0 (s-wave interaction strength) = 0.26
    Fit to T_c^{s-wave}=3 K via eq. (6) with ℏω_D=0.01 eV; controls the s-wave free energy in Fig. 4.
  • VN_0 (p-wave interaction strength) = 0.78
    Fit to T_c^{p-wave}=2.4 K, extrapolated from the high-field branch of B_c2^||(T) in Fig. 2d; controls the transition field B* in the model.
  • ℏω_D (Debye energy) = 0.01 eV
    Chosen as order of the largest phonon frequency (ref 46); enters eq. (6) and thus the U,V estimates.
  • Continuum band parameters (α, m, ε, μ) = α=0.81 eV, m=-0.43 eV^-1, ε=0.63 eV, μ=-2.22 eV
    Chosen so the model reproduces DFT/ARPES bands around Γ (Supplementary Note 2.4); they set the spin splitting used in the gap equations but are not fit to the superconducting data.
assumptions (5)
  • standard math BCS gap equation and Dynes/Maki density-of-states formulas describe the tunnelling conductance.
    Used throughout for extracting Δ, ζ, and Γ from spectra, eqs. (1)-(3), (9)-(12).
  • domain assumption beta-PdBi2 hosts locally broken inversion symmetry with Rashba-like SOC and opposite spin helicities in the two Bi sublayers.
    Central to the model; supported by prior ARPES/QPI work (refs 26-30), not rederived here.
  • domain assumption Orbital depairing suppresses s-wave and p-wave phases similarly, so it can be omitted from the free-energy comparison.
    The paper explicitly states this after Fig. 4b; if false, the computed B* relative to B_c2 could change.
  • ad hoc to paper A local U plus interlayer V interaction, projected onto one s-wave and one spin-polarized triplet channel, captures the competing physics.
    The interaction Hamiltonian (eq. 8) and the two-component order parameter (eq. S20) are chosen as the simplest model that yields the observed transition; they are not derived from microscopic theory.
  • domain assumption The continuum model's circular Fermi surface and neglected warping/interlayer hopping preserve the essential spin structure at the Fermi surface.
    Justified as a simplification in Supplementary Note 2.4 to make gap equations analytically tractable.

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Pith. "Pith review of Multiphase superconductivity in PdBi2." pith.science (2026). https://pith.science/paper/VA2DKIVH

@misc{pith2026241109239,
  author       = {Pith},
  title        = {Pith review of: Multiphase superconductivity in PdBi2},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VA2DKIVH}},
  note         = {Machine review of arXiv:2411.09239}
}
read the original abstract

Unconventional superconductivity, where electron pairing does not involve electron-phonon interactions, is often attributed to magnetic correlations in a material. Well known examples include high-T_c cuprates and uranium-based heavy fermion superconductors. Less explored are unconventional superconductors with strong spin-orbit coupling, where interactions between spin-polarised electrons and external magnetic field can result in multiple superconducting phases and field-induced transitions between them, a rare phenomenon in the superconducting state. Here we report a magnetic-field driven phase transition in \beta-PdBi2, a layered non-magnetic superconductor. Our tunnelling spectroscopy on thin PdBi2 monocrystals incorporated in planar superconductor-insulator-normal metal junctions reveals a marked discontinuity in the superconducting properties with increasing in-plane field, which is consistent with a transition from conventional (s-wave) to nodal pairing. Our theoretical analysis suggests that this phase transition may arise from spin polarisation and spin-momentum locking caused by locally broken inversion symmetry, with p-wave pairing becoming energetically favourable in high fields. Our findings also reconcile earlier predictions of unconventional multigap superconductivity in \beta-PdBi2 with previous experiments where only a single s-wave gap could be detected.

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