Phonon-mediated intrinsic topological superconductivity in Fermi arcs

Bj\"orn Trauzettel; Kristian M{\ae}land; Masoud Bahari

REVIEW 4 major objections 4 minor 2 cited by

Phonon-mediated intrinsic topological superconductivity in Fermi arcs

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Phonons can turn Weyl-semimetal Fermi arcs into topological superconductors.

desk verdict A coherent mechanism paper giving phonon-mediated chiral p-wave pairing on Fermi arcs with a genuinely new gap-suppression signature, but the Majorana conclusion is asserted without a BdG invariant calculation and the quantitative results are parameter-sensitive. read the letter →

arxiv 2506.03250 v2 pith:RX3EHKZ2 submitted 2025-06-03 cond-mat.supr-con cond-mat.mtrl-sci

Kristian M{ae}land , Masoud Bahari , Björn Trauzettel This is my paper

classification cond-mat.supr-concond-mat.mtrl-sci

keywords WeylsemimetalFermiarcstopologicalsuperconductivityphonon-mediatedpairingchiralp-waveMajoranaboundstatesBCStheoryPtBi2

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

This paper argues that the crystal lattice's own vibrations can make the surface electrons of a Weyl semimetal superconducting in a topologically nontrivial state, without any proximity contact or external pairing mechanism. Working in a slab geometry with two surfaces and the bulk in one formalism, the authors find that phonon-mediated attraction acts most strongly on the nondegenerate Fermi-arc surface states, so surface superconductivity dominates over bulk superconductivity in a window of chemical potentials around the Weyl nodes. The resulting gap is spinless chiral $p$-wave, $p_x+ip_y$ in the band basis, the same state that was previously proposed for a superconductor placed on a topological insulator; vortices then host Majorana bound states once time-reversal symmetry is weakly broken. For the chosen material parameters the calculation gives a dimensionless coupling $\lambda\approx 0.38$ and $T_c \approx 19.7$ K, and it predicts a distinctive fingerprint: the gap magnitude is suppressed in the center of the Fermi arc rather than peaked there, because the phonon pairing is nonlocal across layers.

What carries the argument

The engine of the argument is the phonon-mediated electron-electron interaction derived from a Taylor expansion of the hopping integrals, $\nabla_{\bar\delta}t_{\ell\ell'}(\bar\delta)=-\chi\,\bar\delta\,t_{\ell\ell'}(\bar\delta)$, combined with a slab-geometry phonon spectrum computed by a force-constant method. In the band basis this interaction is nonlocal in the layer index: the out-of-plane part does not vanish at zero momentum transfer because the phonon eigenvectors on adjacent layers differ, so low-energy optical phonons at $q=0$ can couple strongly to states with partial bulk penetration. Feeding the symmetrized interaction $\bar{V}_{\mathbf k\mathbf k'}$ into the linearized BCS gap equation yields an eigenvalue problem whose leading eigenvector is the gap function $\Delta_{\mathbf k}\sim k_x+ik_y$ on the Fermi arc, with the pairing strength set by the eigenvector overlap and the Fermi-surface density of states.

What would settle it

Angle-resolved photoemission that maps the superconducting gap along a Fermi arc in a candidate Weyl semimetal such as MoTe$_2$ or TaIrTe$_4$ would settle the claim: the mechanism predicts a fully opened gap whose magnitude dips at the arc center and peaks between the center and the endpoints, with zero-bias conductance peaks at vortex cores in scanning tunneling microscopy. Seeing nodes at the arc center (as reported for PtBi$_2$) or finding no surface-dominated superconductivity above the bulk $T_c$ would contradict the prediction.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that phonon-mediated pairing in the Fermi-arc surface states of a Weyl semimetal intrinsically realizes the topological superconducting state previously proposed for a superconductor placed on a topological insulator surface. Because the Fermi arc is a nondegenerate surface band, Cooper pairs formed there behave like spinless fermions, and spin-orbit coupling imprints a chiral $p_x+ip_y$ momentum dependence onto an otherwise phonon-generated attraction. In the slab calculation the bottom-surface arc dominates the superconductivity for $\mu<0$ down to at least $\mu=-0.1t$, with the top surface and bulk couplings nearly decoupled; the gap on the bottom arc is fully gapped with maxima between the arc center and its endpoints and a local minimum at the center. The suppression at the center is the paper's phonon-specific signature, arising from the out-of-plane component of the electron-phonon coupling that is active when surface states penetrate into the bulk, in contrast to a local Hubbard attraction which would put the maximum gap at the arc center.

Load-bearing premise

The results rest on an assumed form of the electron-phonon coupling, namely that lattice vibrations alter the electron hopping in proportion to the hopping direction with a strength set by hand, together with phonon frequencies tuned to put many low-energy lattice vibrations at zero momentum; if a real material's vibrations couple differently, the predicted central dip in the gap and the high critical temperature would not follow.

Editorial extensions

If this is right

In a slab of a Weyl semimetal, superconductivity can set in on one surface Fermi arc before the bulk or the opposite surface become superconducting, so a measurable surface gap can coexist with a normal bulk over a range of chemical potentials.
The band-basis gap is spinless chiral $p$-wave, and a small out-of-plane magnetic field breaks time-reversal symmetry, turning the surface into a two-dimensional topological superconductor whose vortices host Majorana bound states and whose odd vortex count gives a chiral edge state.
The absolute value of the gap on the dominant Fermi arc is suppressed at the arc center and largest between the center and the endpoints, a momentum-resolved fingerprint that distinguishes phonon pairing from a local Hubbard interaction.
Top and bottom surface Fermi arcs behave almost independently and can have different critical temperatures (about 19.7 K and 9.2 K for the chosen parameters), effectively two independent two-dimensional topological superconductors under a weak field.
Candidate Weyl semimetals with lighter atoms, such as MoTe$_2$ or TaIrTe$_4$, may show the predicted fully gapped arc with a central dip, whereas existing angle-resolved photoemission evidence of nodes in PtBi$_2$ points to a different or additional pairing mechanism.

Reading between the lines

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

Surface termination or strain, by changing how far Fermi-arc states penetrate into the bulk, should be able to tune the strength of the central gap dip, because the out-of-plane phonon coupling is what creates it.
The near decoupling of top surface, bottom surface, and bulk superconductivity implies that a sample with slightly different surfaces could display two distinct surface critical temperatures, each independently tunable by chemical potential.
Because the central dip disappears when the phonon parameters are changed (the paper reports only about 2% suppression for an alternative set), the dip is a sensitive fingerprint of the assumed low-energy optical-phonon spectrum, not a generic property of phonon pairing.
A self-consistent or strong-coupling treatment beyond the linearized BCS equation would likely shift the quantitative $T_c$ values, but the symmetry of the gap and the surface-over-bulk hierarchy are the assertions most worth testing experimentally.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Kristian,

My take on Mæland et al. (arXiv:2506.03250): the genuinely new thing is that phonons, not a local Hubbard attraction, can produce spinless chiral p-wave pairing on Weyl-semimetal Fermi arcs, and that the nonlocal EPC gives a dip in |Δ| at the arc center. That suppression is a real contrast with the s-wave result of Trama et al., and the derivation is coherent. The paper also does the right thing in naming falsifiable signatures: gap suppression, a surface T_c hierarchy, and vortex zero-bias peaks. It is honest about PtBi2, where recent ARPES shows a nodal gap, and points instead to MoTe2 and TaIrTe4.

The soft spots, in proportion. The biggest one is the topological claim. The gap in the band basis looks p_x+ip_y, and the paper invokes Fu-Kane to say Majorana bound states appear in vortex cores after a small Zeeman field. But the Fermi arc is an open Fermi surface, not the closed surface-state Fermi surface of the TI case, and the bulk Weyl nodes remain gapless. The paper never computes a BdG Chern number or solves the vortex problem. The gauge dependence discussion in Appendix D.1 makes clear the gap phase in the band basis is a gauge choice, and the assertion that the topological classification is gauge invariant is not demonstrated. That chain is asserted rather than shown. I would not say the mechanism is wrong, but the Majorana conclusion is overreach as written.

Second, the quantitative claims are fragile. The headline λ≈0.38, T_c≈19.7 K and the pronounced arc-center suppression come from phonon parameters and an EPC ansatz (∇δ t = -χ δ t) chosen to produce them. The authors' own Appendix E shows that an alternative reasonable phonon set gives λ≈0.10, T_c≈0.016 K, and only 2% suppression. This is a mechanism proposal, not a quantitative prediction, and they are candid about it, but the abstract's concrete numbers will likely be taken more seriously than they deserve.

Third, no code or data is provided, and the numerics are a substantial part of the work. Minor for a theory paper, but it would help.

Who should read it: theorists working on surface superconductivity in Weyl semimetals and intrinsic topological superconductivity; experimentalists may use the falsifiable signatures. The work deserves a serious referee. I would send it to review with a request to either strengthen the topological argument (compute the class-D invariant for the coupled surface-bulk system, or at minimum show why the open arc can be treated as a gapped 2D topological superconductor) or soften the Majorana claim. That is a major-revision path, not a rejection.

Yours, [Name]

Referee Report

4 major / 4 minor

Summary. The paper studies phonon-mediated superconductivity on the Fermi arcs of a slab of a hexagonal Weyl semimetal. Starting from an effective two-orbital electron model with spin-orbit coupling and broken inversion symmetry, the authors derive a force-constant phonon spectrum for the slab, construct an electron-phonon coupling by Taylor-expanding the hopping (Sec. IV), and obtain an effective electron-electron interaction via a Schrieffer-Wolff transformation. Solving the linearized, Fermi-surface-averaged BCS gap equation (Eq. 5), they find that the leading instability is localized on the bottom-surface Fermi arc and has a chiral p-wave form, with a gap suppression in the center of the arc. For the chosen parameters they report lambda ~ 0.38 and Tc ~ 19.7 K, with the surface state dominating over the bulk for a range of chemical potentials around the Weyl nodes. The paper interprets this as an intrinsic realization of the Fu-Kane topological superconductor and predicts Majorana bound states in vortex cores after a weak out-of-plane magnetic field. An appendix discusses the competition between surface and bulk superconductivity, the gauge dependence of the band-basis gap, and the sensitivity of the results to phonon parameters.

Significance. If fully established, the paper would provide a concrete phonon-only mechanism for intrinsic topological superconductivity on Weyl-semimetal surfaces, going beyond the usual local Hubbard attraction and making a falsifiable experimental prediction (a full superconducting gap with a central suppression, and vortex-core zero-bias peaks). The derivation is coherent and has real strengths: the chiral p-wave symmetry is an emergent solution of the gap equation rather than an input; the calculation is also performed on the full Fermi surface and for a second phonon parameter set; and the authors explicitly compare the gap symmetry in the original orbital basis with an earlier symmetry analysis. The main limitations are, first, that the topological conclusions rest on the momentum-space phase of a band-basis gap rather than on a BdG invariant, and second, that several quantitative headline results are obtained with parameter values that are, by the authors' own account, tuned. The paper is honest about these issues in the appendices, but the main text and abstract do not always carry the same caveats.

major comments (4)

[Sec. V, Sec. VI, Appendix D.1] The abstract's claim that the gapped Fermi arcs lead to Majorana bound states in vortex cores is not derived within the manuscript. The calculation stops at the linearized gap equation (Eq. 5) and the observation that the gap on the bottom arc has the form k_x + i k_y. A Fermi arc is an open segment of the Fermi surface embedded in a gapless 3D Weyl semimetal, not a closed 2D Fermi surface, so the standard Chern-number argument for a fully gapped 2D chiral p-wave superconductor does not automatically apply. Moreover, Appendix D.1 explicitly states that the band-basis gap can appear as p_x+ip_y or p_x-ip_y depending on which eigenvector component is fixed to be real and positive, and that multiplying the eigenvector by a momentum-dependent phase factor changes the winding. The paper asserts that the topological classification is gauge invariant, but it never evaluates a BdG invariant (for instance, the class-D Chern number after adding a small Zeeman term) or solves the BdG equations with a vortex. Without such a calculation, 'leading to Majorana bound states' is an extrapolation, not a demonstrated consequence. Please either add the BdG calculation or clearly state this part as a conjecture.
[Sec. IV, Sec. V, Appendix E.5, Fig. 7] The quantitative claims are strongly dependent on hand-picked parameters, and this dependence is acknowledged in the text. The electron-phonon coupling uses the ansatz nabla_delta t = -chi delta t with chi = 8 chosen 'larger' than in Ref. [60], and Sec. V says the phonon parameters are tuned 'to ensure many such modes'. Appendix E.5 and Fig. 7 show that with the alternative phonon set gamma_3 = gamma_6 = gamma_1, the central suppression shrinks to about 2% and the coupling and critical temperature drop to lambda ~ 0.10 and Tc ~ 0.016 K. Thus the pronounced suppression in Fig. 3(d) and the headline values lambda ~ 0.38 and Tc ~ 19.7 K are properties of one parameter choice, not robust predictions of the model. The authors do acknowledge this in the appendix, but the abstract and the main-text discussion present the suppression as a general consequence of the nonlocal origin of electron-phonon coupling. Please quantify the parameter region in which the suppression is observable, or reframe the quantitative statements as a proof-of-principle demonstration.
[Sec. III, Sec. V] The slab 'optical' phonon modes invoked for the suppression mechanism are not true optical branches of the material. The effective model has a single atomic basis, so with three-dimensional periodic boundary conditions it has only three acoustic phonon branches. The 3L-3 modes with nonzero energy at zero in-plane momentum in the slab are standing-wave solutions of the acoustic branches quantized by the open boundary condition, not the physical optical phonons of the nine-atom PtBi2 basis. The suppression of |Delta| in the center of the arc is attributed to low-energy optical-type phonons at q = 0 (Sec. V), which in this model are finite-size slab modes whose spectrum depends on L and on the chosen force constants. The authors note that the effective model 'catches the three acoustic modes', which makes the later reliance on low-energy optical modes at q = 0 potentially inconsistent. Please clarify whether such slab-confined modes are expected in the candidate materials, or state explicitly that the suppression is a property of the effective slab model rather than of PtBi2-like materials.
[Appendix D.1] The gauge-fixing procedure introduces an arbitrary convention into the computed phase of Delta(k). The authors find that random local gauges are 'problematic' and therefore set one element of each electron eigenvector to be real and positive; switching from spin-up to spin-down changes the gap from p_x+ip_y to p_x-ip_y. The absolute value of the gap is gauge invariant, as are physical observables, but the phase winding that is used to infer chiral p-wave pairing is not fixed by the calculation unless a physical gauge is specified. The statement that the topological classification is gauge invariant would be unproblematic if an explicit BdG invariant were computed, but in the absence of such an invariant the p_x+ip_y winding in the band basis remains a convention-dependent property. Please provide an explicit gauge-invariant quantity, or discuss which physical condition selects the reported gauge.

minor comments (4)

[Sec. V] There is a duplicated word in the sentence 'when only focusing the the bottom surface Fermi arc'; it should be 'focusing on the bottom surface Fermi arc'.
[Fig. 3 caption] The values lambda ~ 0.38 and Tc ~ 19.7 K appear only in the caption and are conditional on t = 1 eV. They should be stated in the main text with an explicit caveat that they are parameter-dependent estimates.
[Appendix C] The statement that chi = 8 is chosen 'larger' than in Ref. [60] would benefit from a quantitative justification; the two calculations use different orbital models, so a direct comparison of chi values is not self-evident.
[Appendix E.5] The scaling lambda ~ chi^2/M is asserted without derivation; a brief derivation or a reference would make the sensitivity analysis easier to follow.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: the chiral p-wave gap is a solution of the linearized gap equation, not an input; self-cited methodology and hand-tuned phonon parameters do not reduce the central result to its assumptions.

full rationale

The paper's central claim—that phonons mediate an odd-parity chiral pairing on the Fermi arc—is obtained by solving the linearized BCS gap equation, Eq. (5), with a phonon-mediated interaction derived from a Taylor-expanded hopping term; the p_x+ip_y form is not inserted as the pairing ansatz. The interaction inherits its chiral structure from the SOC-containing electron eigenvectors (App. E), and the gap eigenvalue problem then selects the odd-parity eigenvector. The suppression of |Delta| in the arc center is a computed consequence of the nonlocal, layer-dependent EPC; although the phonon parameters are chosen to emphasize low-energy optical modes at q=0 and chi=8 is chosen from a cited range, this is parameter choice rather than fitting the target result, and the paper transparently shows an alternative phonon set reduces the suppression to about 2 percent (Fig. 7). The self-citations (Refs. [48,66] for the BCS framework, Ref. [61] for the chi ansatz) are methodological and not load-bearing: the gap equation is standard BCS/Schrieffer-Wolff theory, and the ansatz is explicitly introduced as a model. The remaining critique—that the Majorana claim is inferred from the local gap symmetry without a BdG Chern number or vortex calculation, and that the gap's p_x+ip_y versus p_x-ip_y form is gauge-dependent (App. D.1)—concerns underdemonstration of the topological conclusion, not a reduction of the result to its inputs. No equation in the paper is equivalent by construction to a fitted target, so no circular step is identified.

Assumptions & free parameters 5 free parameters · 8 assumptions · 0 invented entities

The electron model parameters (t_o/t=1.5, β/t=-1.5, μ/t=-0.05, μ_o/t=0.2, α/t=-0.18, γ/t=-0.2) are inherited from the effective PtBi2 model of Ref. [24] and are treated here as inputs, not as parameters introduced by this paper. The three free parameters explicitly introduced by this paper for its mechanism are the phonon force constants γ1, γ3, γ6 (with the additional simplification ratios) and the EPC scale χ. These are not derived from first principles and the quantitative predictions (T_c, suppression strength) depend on them. The central qualitative claim, chiral p-wave pairing and surface dominance, appears robust across phonon parameter sets, but the magnitude of the novel gap-suppression effect is not.

free parameters (5)

Phonon force constant γ1 = γ1 = -(0.005t)^2, t=1 eV
Phenomenological parameter setting the phonon energy scale; chosen to place the phonon DOS peak near 15 meV. Directly affects T_c and the magnitude of the predicted gap suppression.
Phonon force constant ratio γ3/γ1 = 0.45
Chosen by hand; together with γ6 controls the number and dispersion of low-energy optical phonons at q=0, which underpin the out-of-plane EPC effect responsible for the gap suppression.
Phonon force constant ratio γ6/γ1 = 1.5
Chosen by hand for a reasonable phonon spectrum; the authors state the parameters are tuned 'to ensure many such modes' in Fig. 2(a) to demonstrate the suppression effect.
Phonon model simplification ratios (ρ_i=-γ1/(10√2), γ4=γ1, γ5=γ3, γ7=γ6) = as listed
Ad hoc reduction of the 12-parameter force-constant model to three parameters; affects phonon eigenvectors entering the EPC.
Electron-phonon coupling scale χ = 8
Dimensionless constant in the ansatz ∇_δ t = -χ δ t; sets the overall pairing strength. Chosen 'larger' than Ref. [60] based on the assumption of small orbital spread; controls λ and T_c (λ ∝ χ^2/M).

assumptions (8)

standard math Harmonic (second-order) Taylor expansion of the lattice potential describes the phonon dynamics of the slab.
Invoked in Sec. III and App. B; the stability and small-displacement assumptions are standard in force-constant models.
standard math Schrieffer-Wolff transformation eliminates phonons to second order, giving an effective static electron-electron interaction.
Used in Sec. V and App. D; standard in phonon-mediated superconductivity.
standard math The leading superconducting instability is captured by the linearized BCS gap equation on the Fermi surface.
Eq. (5); standard weak-coupling approach.
domain assumption The effective two-orbital hexagonal model of PtBi2 (Ref. [24]) captures the relevant Fermi arc states and Weyl nodes.
Sec. II; the paper's results depend on this model's surface-state wave functions and Fermi surface geometry.
ad hoc to paper Electron-phonon coupling is obtained by Taylor-expanding nearest-neighbor hopping with ∇_δ t = -χ δ t.
Sec. IV and App. C; determines the momentum/layer structure of the pairing interaction, including the out-of-plane coupling responsible for the gap suppression.
ad hoc to paper Only the single nondegenerate band crossing the Fermi level participates in pairing; other bands are neglected.
Sec. IV: 'We have dropped the band index on the electron band operators d_k, focusing only on the band that has an FS.' This is necessary for the spinless p-wave picture.
domain assumption A small out-of-plane magnetic field breaks TRS without changing the Fermi arcs or superconductivity significantly.
Sec. V and App. E4; the Majorana-bound-state claim relies on entering class D this way.
domain assumption The single-atom-basis phonon model captures the relevant acoustic phonons, while the 24 optical branches of the real PtBi2 unit cell are neglected.
Sec. III: 'within the effective model we are catching the three acoustic modes, which given their lower energy should be expected to make the main contribution to superconductivity.'

how reviews work

0 comments

Cite this review

Pith. "Pith review of Phonon-mediated intrinsic topological superconductivity in Fermi arcs." pith.science (2026). https://pith.science/paper/RX3EHKZ2

@misc{pith2026250603250,
  author       = {Pith},
  title        = {Pith review of: Phonon-mediated intrinsic topological superconductivity in Fermi arcs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RX3EHKZ2}},
  note         = {Machine review of arXiv:2506.03250}
}

abstract

We propose that phonons can intrinsically mediate topological superconductivity on the surface of Weyl semimetals. Weyl semimetals are gapless topological materials with nondegenerate zero energy surface states known as Fermi arcs. We derive the phonon spectrum and electron-phonon coupling in an effective model of a Weyl semimetal and apply weak-coupling Bardeen-Cooper-Schrieffer theory of superconductivity. In a slab geometry, we find that surface superconductivity dominates over bulk superconductivity in a range of chemical potentials around the Weyl nodes. The superconducting gap function realizes spinless chiral $p$-wave Cooper pairing in the Fermi arcs, leading to Majorana bound states in the core of vortices. Furthermore, we find a suppression of the absolute value of the gap in the center of the Fermi arcs, which is not captured by a local Hubbard attraction. The suppression is due to the nonlocal origin of electron-phonon coupling, leading to a layer dependence which has important consequences for topological surface states.

Figures

Figures reproduced from arXiv: 2506.03250 by the authors.

**Figure 2.** FIG. 2. (a) Black lines show the phonon spectrum in a slab geometry. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** FIG. 4. Real (a), imaginary (b), and absolute value (c) of the gap function on the full Fermi surface. The values are scaled by the maximum [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: FIG. 5. Real (a), imaginary (b), and absolute value (c) of the gap function on the top surface Fermi arc. The values are scaled by the maximum [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: FIG. 6. Black lines show phonon modes in the slab geometry. Col [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: FIG. 7. Real (a), imaginary (b), and absolute value (c) of the gap function on the bottom surface Fermi arc. The values are scaled by the [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

Discussion (0). Sign in to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

North-South Asymmetry of the Solar Activity at Different Spatial Scales
astro-ph.SR 2025-08 unverdicted novelty 4.0 of 10

Analyzing sunspots and the large-scale surface magnetic field, the paper argues that long-term north-south asymmetric structures come from the mean-field dynamo, while short-term sunspot production is a separate near-...
Quantum Algorithm Software for Condensed Matter Physics
cond-mat.str-el 2025-06 reject novelty 2.0 of 10

A review of quantum algorithm software that advertises a benchmark suite, yet the body contains no benchmarks, data, or code.

Reference graph

Works this paper leans on

99 extracted references · 43 canonical work pages · cited by 2 Pith papers

[60]

Mæland, S

K. Mæland, S. Abnar, J. Benestad, and A. Sudbø, Topologi- cal superconductivity mediated by magnons of helical magnetic states, Phys. Rev. B108, 224515 (2023)

2023
[1]

There aredrphonon modes, wheredis the dimensionality andris the number of atoms in the basis

General framework Phonon dispersions can be derived in specific lattice sys- tems using the system’s symmetries through a force constant approach [57–61]. There aredrphonon modes, wheredis the dimensionality andris the number of atoms in the basis. dof the phonon modes are acoustic, with zero energy at zero momentum, while the remainingd(r−1)modes are opt...
[2]

cos(ky) + 2 cos √ 3kx 2 ! cos ky 2 # ,(A6) gk =t o

Those parameters giveλ≈0.38andT c ≈19.7K ift= 1eV . number inversely proportional to the standard deviation of the atomic orbitals [60, 61]. Appendix C provides more details of the derivation of the EPC. V . SUPERCONDUCTIVITY A Schrieffer-Wolff transformation [64, 65] yields an effec- tive electron-electron interaction mediated by the phonons. Focusing on...
[3]

There are three mirror planes meeting at120 ◦ normal to thexyplane, with the inter- section being thezaxis

Hexagonal lattice In our case, we consider a 3D crystal with a one-atomic ba- sis (r= 1andα, βindices become superfluous) with symme- tries given by the P31m space group. There are three mirror planes meeting at120 ◦ normal to thexyplane, with the inter- section being thezaxis. Also, there is a three-fold rotational symmetry about thezaxis. For rotations ...
[4]

[62, 63] for other studies of phonons with open boundary conditions, which includes examples of the range of phonon energies for certain materials

Phonon spectrum with one open boundary condition See Refs. [62, 63] for other studies of phonons with open boundary conditions, which includes examples of the range of phonon energies for certain materials. Reference [63] also discusses surface phonons and enhanced EPC at the surface. The origin is that interatomic distances may change close to the surfac...
[5]

The combinationsg m kk′gm −k,−k′ and gm k,−k′gm −k,k′ both contain productse ziµ qmez′ iν −q,m

Gauge dependence Since it is ¯V FS kk′ =− P m gm kk′gm −k,−k′/ωk−k′,m +P m gm k,−k′gm −k,k′/ωk+k′,m which enters in the gap equation we focus on its behavior. The combinationsg m kk′gm −k,−k′ and gm k,−k′gm −k,k′ both contain productse ziµ qmez′ iν −q,m. The phonon eigenvectors have the propertyˆe −q,m = [ˆeqm]∗ which can be shown from the fact that the d...
[6]

Hence, they correspond to moving all ions the same amount inx, y, orzdirection

Zero momentum transfer The three phonon modes with zero energy atq= 0in the slab geometry have eigenvectors with elements1/Lat the x, y, orzposition at each layer and otherwise zero. Hence, they correspond to moving all ions the same amount inx, y, orzdirection. So, both for the in-plane and out-of-plane type EPC, the numerator inV FS kk′ goes to zero whe...
[7]

In both cases, we see that the cou- pling is negligible fork ′ in the bulk and top surface parts of the FS, supporting the claim in Sec

Bulk and surface separability and origin of gap symmetry Figures 4(d)-4(g) show ¯V FS kk′ as a function ofk ′ on the full FS withkfixed at the center of the Fermi arc on bottom sur- face and halfway between the center and endpoint of the bot- tom surface Fermi arc. In both cases, we see that the cou- pling is negligible fork ′ in the bulk and top surface ...

Show all 99 references

[8]

We can transform our gap function to the original basis and compare, usingd † k =P ziℓσ vkziℓσc† kziℓσ

Superconducting gap in original basis Reference [24] performed a symmetry analysis in the spin basis of possible superconducting pairing in Fermi arcs in the model we have adopted. We can transform our gap function to the original basis and compare, usingd † k =P ziℓσ vkziℓσc†...
[9]

The gap on the bottom surface is the same as shown in Fig

Competition of surface and bulk superconductivity Figures 4(a)-4(c) show the superconducting gap function on the full FS. The gap on the bottom surface is the same as shown in Fig. 3 when considering only the bottom surface Fermi arc. Meanwhile, the gap in the bulk and top sur...
[10]

This scenario is disad- vantageous for superconductivity

Consequence of breaking time-reversal symmetry It is worth noting that in a Weyl semimetal with broken inversion symmetry, additionally breaking time-reversal sym- metry means there is no guarantee that for a Fermi arc atk there is a corresponding one at−k. This scenario is di...
[11]

Sensitivity to phonon properties Here we show that the main results are robust towards changing the phonon properties. In Fig. 6, we compare the phonon spectrum with two different sets of phonon parame- ters, namely, those used in the rest of the work to emphasize the effect o...
[12]

Yan and C

B. Yan and C. Felser, Topological Materials: Weyl Semimetals, Annu. Rev. Condens. Matter Phys.8, 337 (2017)

2017
[13]

N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys.90, 015001 (2018)

2018
[14]

Kuibarov, O

A. Kuibarov, O. Suvorov, R. V ocaturo, A. Fedorov, R. Lou, L. Merkwitz, V . V oroshnin, J. I. Facio, K. Koepernik, A. Yaresko, G. Shipunov, S. Aswartham, J. v. d. Brink, B. B ¨uchner, and S. Borisenko, Evidence of superconducting Fermi arcs, Nature626, 294 (2024)

2024
[15]

Schimmel, Y

S. Schimmel, Y . Fasano, S. Hoffmann, J. Besproswanny, L. T. Corredor Bohorquez, J. Puig, B.-C. Elshalem, B. Kalisky, G. Shipunov, D. Baumann, S. Aswartham, B. B ¨uchner, and C. Hess, Surface superconductivity in the topological Weyl semimetal t-PtBi2, Nat. Commun.15, 9895 (20...

2024
[16]

Hoffmann, S

S. Hoffmann, S. Schimmel, R. V ocaturo, J. Puig, G. Shipunov, O. Janson, S. Aswartham, D. Baumann, B. B¨uchner, J. van den Brink, Y . Fasano, J. I. Facio, and C. Hess, Fermi Arcs Dominat- ing the Electronic Surface Properties of Trigonal PtBi 2, Adv. Phys. Res.4, 2400150 (2024)

2024
[17]

Veyrat, V

A. Veyrat, V . Labracherie, D. L. Bashlakov, F. Caglieris, J. I. Facio, G. Shipunov, T. Charvin, R. Acharya, Y . Naidyuk, R. Gi- raud, J. van den Brink, B. B¨uchner, C. Hess, S. Aswartham, and J. Dufouleur, Berezinskii–Kosterlitz–Thouless Transition in the Type-I Weyl Semimeta...

2023
[18]

Huang, L

X. Huang, L. Zhao, S. Schimmel, J. Besproswanny, P. H ¨artl, C. Hess, B. B ¨uchner, and M. Bode, Sizable superconducting gap and anisotropic chiral topological superconductivity in the Weyl semimetal PtBi2, arXiv:2507.13843 (2025)

2025 arXiv
[19]

J. A. Moreno, P. G. Talavera, E. Herrera, S. L. Valle, Z. Li, L.-L. Wang, S. Bud’ko, A. I. Buzdin, I. Guillam ´on, P. C. Can- field, and H. Suderow, Robust surface superconductivity and vortex lattice in the Weyl semimetalγ-PtBi2, arXiv:2508.04867 (2025)

2025
[20]

Naidyuk, O

Y . Naidyuk, O. Kvitnitskaya, D. Bashlakov, S. Aswartham, I. Morozov, I. Chernyavskii, G. Fuchs, S.-L. Drechsler, R. H ¨uhne, K. Nielsch, B. B ¨uchner, and D. Efremov, Surface superconductivity in the Weyl semimetal MoTe 2 detected by point contact spectroscopy, 2D Mater.5, 04...

2018
[21]

Y . Xing, Z. Shao, J. Ge, J. Luo, J. Wang, Z. Zhu, J. Liu, Y . Wang, Z. Zhao, J. Yan, D. Mandrus, B. Yan, X.-J. Liu, M. Pan, and J. Wang, Surface superconductivity in the type II Weyl semimetal TaIrTe4, Natl. Sci. Rev.7, 579 (2020)

2020
[22]

G. Y . Cho, J. H. Bardarson, Y .-M. Lu, and J. E. Moore, Su- perconductivity of doped Weyl semimetals: Finite-momentum pairing and electronic analog of the 3He-Aphase, Phys. Rev. B 86, 214514 (2012)

2012
[23]

Wei, S.-P

H. Wei, S.-P. Chao, and V . Aji, Odd-parity superconductivity in Weyl semimetals, Phys. Rev. B89, 014506 (2014)

2014
[24]

Hosur, X

P. Hosur, X. Dai, Z. Fang, and X.-L. Qi, Time-reversal-invariant topological superconductivity in doped Weyl semimetals, Phys. Rev. B90, 045130 (2014)

2014
[25]

Bednik, A

G. Bednik, A. A. Zyuzin, and A. A. Burkov, Superconductivity in Weyl metals, Phys. Rev. B92, 035153 (2015)

2015
[26]

Alidoust, K

M. Alidoust, K. Halterman, and A. A. Zyuzin, Superconductiv- ity in type-II Weyl semimetals, Phys. Rev. B95, 155124 (2017)

2017
[27]

Y . Qi, P. G. Naumov, M. N. Ali, C. R. Rajamathi, W. Schnelle, O. Barkalov, M. Hanfland, S.-C. Wu, C. Shekhar, Y . Sun, V . S¨uß, M. Schmidt, U. Schwarz, E. Pippel, P. Werner, R. Hille- brand, T. F¨orster, E. Kampert, S. Parkin, R. J. Cava, C. Felser, B. Yan, and S. A. Medvede...

2016
[28]

M. R. van Delft, S. Pezzini, M. K ¨onig, P. Tinnemans, N. E. Hussey, and S. Wiedmann, Two- and Three-Dimensional Su- perconducting Phases in the Weyl Semimetal TaP at Ambient Pressure, Crystals10, 288 (2020)

2020
[29]

Shipunov, I

G. Shipunov, I. Kovalchuk, B. R. Piening, V . Labracherie, A. Veyrat, D. Wolf, A. Lubk, S. Subakti, R. Giraud, J. Du- fouleur, S. Shokri, F. Caglieris, C. Hess, D. V . Efremov, B. B ¨uchner, and S. Aswartham, PolymorphicPtBi 2: Growth, structure, and superconducting properties...

2020
[30]

D. L. Bashlakov, O. E. Kvitnitskaya, G. Shipunov, S. Aswartham, O. D. Feya, D. V . Efremov, B. B ¨uchner, and Yu. G. Naidyuk, Electron-phonon interaction and point contact enhanced superconductivity in trigonal PtBi 2, Low Temp. Phys.48, 747 (2022)

2022
[31]

Zabala, V

J. Zabala, V . F. Correa, F. J. Castro, and P. Pedrazzini, Enhanced weak superconductivity in trigonalγ-PtBi 2, J. Phys.: Condens. Matter36, 285701 (2024)

2024
[32]

Nomani and P

A. Nomani and P. Hosur, Intrinsic surface superconducting in- stability in type-I Weyl semimetals, Phys. Rev. B108, 165144 (2023)

2023
[33]

X. Bai, W. LiMing, and T. Zhou, Superconductivity in Weyl semimetals with time reversal symmetry, New J. Phys.27, 013003 (2025)

2025
[34]

Trama, V

M. Trama, V . K ¨onye, I. C. Fulga, and J. van den Brink, Self- consistent surface superconductivity in time-reversal symmetric Weyl semimetals, Phys. Rev. B112, 064514 (2025). 14

2025
[35]

V ocaturo, K

R. V ocaturo, K. Koepernik, J. I. Facio, C. Timm, I. C. Fulga, O. Janson, and J. van den Brink, Electronic structure of the surface-superconducting Weyl semimetalPtBi 2, Phys. Rev. B 110, 054504 (2024)

2024
[36]

H. Waje, F. Jakubczyk, J. v. d. Brink, and C. Timm, Ginzburg- Landau theory for unconventional surface superconductivity in PtBi2, arXiv:2507.02415 (2025)

2025
[37]

Nayak, S

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys.80, 1083 (2008)

2008
[38]

Leijnse and K

M. Leijnse and K. Flensberg, Introduction to topological super- conductivity and Majorana fermions, Semicond. Sci. Technol. 27, 124003 (2012)

2012
[39]

B. A. Bernevig and T. L. Hughes,Topological Insulators and Topological Superconductors(Princeton University Press, Princeton, NJ, 2013)

2013
[40]

Sato and Y

M. Sato and Y . Ando, Topological superconductors: a review, Rep. Prog. Phys.80, 076501 (2017)

2017
[41]

A. Y . Kitaev, Unpaired Majorana fermions in quantum wires, Phys.-Usp.44, 131 (2001)

2001
[42]

Frolov, Quantum computing’s reproducibility crisis: Majo- rana fermions, Nature592, 350 (2021)

S. Frolov, Quantum computing’s reproducibility crisis: Majo- rana fermions, Nature592, 350 (2021)

2021
[43]

Fu and C

L. Fu and C. L. Kane, Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator, Phys. Rev. Lett.100, 096407 (2008)

2008
[44]

Fukui and T

T. Fukui and T. Fujiwara, Topological Stability of Majorana Zero Modes in Superconductor–Topological Insulator Systems, J. Phys. Soc. Jpn.79, 033701 (2010)

2010
[45]

Y . Oreg, G. Refael, and F. von Oppen, Helical Liquids and Ma- jorana Bound States in Quantum Wires, Phys. Rev. Lett.105, 177002 (2010)

2010
[46]

Nakosai, Y

S. Nakosai, Y . Tanaka, and N. Nagaosa, Two-dimensionalp- wave superconducting states with magnetic moments on a con- ventionals-wave superconductor, Phys. Rev. B88, 180503(R) (2013)

2013
[47]

A. O. Zlotnikov, M. S. Shustin, and A. D. Fedoseev, Aspects of Topological Superconductivity in 2D Systems: Noncollinear Magnetism, Skyrmions, and Higher-order Topology, J. Super- cond. Nov. Magn.34, 3053 (2021)

2021
[48]

P. M. R. Brydon, S. Das Sarma, H.-Y . Hui, and J. D. Sau, Odd- parity superconductivity from phonon-mediated pairing: Appli- cation toCu xBi2Se3, Phys. Rev. B90, 184512 (2014)

2014
[49]

Zhang and S

R.-X. Zhang and S. Das Sarma, Intrinsic Time-Reversal- Invariant Topological Superconductivity in Thin Films of Iron- Based Superconductors, Phys. Rev. Lett.126, 137001 (2021)

2021
[50]

F. O. von Rohr, Chemical Principles of Intrinsic Topological Superconductors, Chem. Mater.35, 9455 (2023)

2023
[51]

J. Kim, K. M. Fijalkowski, J. Kleinlein, C. Schumacher, A. Markou, C. Gould, S. Schreyeck, C. Felser, and L. W. Molenkamp, Molecular beam epitaxy of a half-Heusler topo- logical superconductor candidate YPtBi, Phys. Rev. Mater.7, 024802 (2023)

2023
[52]

Bahari, S.-B

M. Bahari, S.-B. Zhang, C.-A. Li, S.-J. Choi, P. R ¨ußmann, C. Timm, and B. Trauzettel, Helical Topological Superconduct- ing Pairing at Finite Excitation Energies, Phys. Rev. Lett.132, 266201 (2024)

2024
[53]

M. S. Scheurer, Mechanism, time-reversal symmetry, and topology of superconductivity in noncentrosymmetric systems, Phys. Rev. B93, 174509 (2016)

2016
[54]

Li, L.-H

S. Li, L.-H. Hu, R.-X. Zhang, and S. Okamoto, Topological superconductivity from forward phonon scatterings, Commun. Phys.6, 235 (2023)

2023
[55]

L. M. Cangemi, A. S. Mishchenko, N. Nagaosa, V . Cataudella, and G. De Filippis, Topological Quantum Transition Driven by Charge-Phonon Coupling in the Haldane Chern Insulator, Phys. Rev. Lett.123, 046401 (2019)

2019
[56]

Islam, K

M. Islam, K. Bhattacharyya, and S. Basu, Electron-phonon cou- pling induced topological phase transition in anα−T3 Haldane- Holstein model, Phys. Rev. B110, 045426 (2024)

2024
[57]

Bhattacharyya, S

K. Bhattacharyya, S. Lahiri, M. Islam, and S. Basu, Holstein polaron in a pseudospin-1 quantum spin Hall system: First- and second-order topological phase transitions, Phys. Rev. B110, 235432 (2024)

2024
[58]

Lahiri, K

S. Lahiri, K. Bhattacharyya, and S. Basu, Emergent topological phases and coexistence of gapless and spectral-localized Flo- quet quantum spin Hall states via electron-phonon interaction, Phys. Rev. B112, 115406 (2025)

2025
[59]

Mæland and A

K. Mæland and A. Sudbø, Topological Superconductivity Me- diated by Skyrmionic Magnons, Phys. Rev. Lett.130, 156002 (2023)

2023
[61]

Vi ˜nas Bostr ¨om and E

F. Vi ˜nas Bostr ¨om and E. Vi ˜nas Bostr ¨om, Magnon-mediated topological superconductivity in a quantum wire, Phys. Rev. Res.6, L022042 (2024)

2024
[62]

C. Sun, K. Mæland, and A. Sudbø, Stability of superconducting gap symmetries arising from antiferromagnetic magnons, Phys. Rev. B108, 054520 (2023)

2023
[63]

C. Sun, K. Mæland, E. Thingstad, and A. Sudbø, Strong- coupling approach to temperature dependence of competing orders of superconductivity: Possible time-reversal symmetry breaking and nontrivial topology, Phys. Rev. B109, 174520 (2024)

2024
[64]

Thingstad, J

E. Thingstad, J. Hutchinson, D. Loss, and J. Klinovaja, Topo- logical interlayer superconductivity in a van der Waals het- erostructure, Phys. Rev. B111, L060505 (2025)

2025
[65]

S. D. Lundemo and A. Sudbø, Topological superconductivity induced by a Kitaev spin liquid, Phys. Rev. B109, 184508 (2024)

2024
[66]

Bardeen, L

J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Microscopic The- ory of Superconductivity, Phys. Rev.106, 162 (1957)

1957
[67]

Bardeen, L

J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of Super- conductivity, Phys. Rev.108, 1175 (1957)

1957
[68]

Bruus and K

H. Bruus and K. Flensberg,Many-Body Quantum Theory in Condensed Matter Physics: An Introduction(Oxford Univer- sity Press, Oxford, 2004)

2004
[69]

J. N. Kløgetvedt, Topological Magnon-Phonon Hybrid Excita- tions and Hall Effects in Two-Dimensional Ferromagnets, Mas- ter thesis, Norwegian University of Science and Technology, https://hdl.handle.net/11250/3097131 (2023)

2023
[70]

Sylju ˚asen, Transverse quantum transport in multiband Bose- systems, Master thesis, Norwegian University of Science and Technology, https://hdl.handle.net/11250/3155941 (2024)

E. Sylju ˚asen, Transverse quantum transport in multiband Bose- systems, Master thesis, Norwegian University of Science and Technology, https://hdl.handle.net/11250/3155941 (2024)

2024
[71]

Thingstad, A

E. Thingstad, A. Kamra, J. W. Wells, and A. Sudbø, Phonon- mediated superconductivity in doped monolayer materials, Phys. Rev. B101, 214513 (2020)

2020
[72]

Leraand, K

K. Leraand, K. Mæland, and A. Sudbø, Phonon-mediated spin- polarized superconductivity in altermagnets, arXiv:2502.08704 (2025)

2025
[73]

A. A. Lucas, Phonon Modes of an Ionic Crystal Slab, J. Chem. Phys.48, 3156 (1968)

1968
[74]

Benedek, M

G. Benedek, M. Bernasconi, V . Chis, E. Chulkov, P. M. Echenique, B. Hellsing, and J. P. Toennies, Theory of surface phonons at metal surfaces: recent advances, J. Phys.: Condens. Matter22, 084020 (2010)

2010
[75]

Bardeen and D

J. Bardeen and D. Pines, Electron-Phonon Interaction in Metals, Phys. Rev.99, 1140 (1955). 15

1955
[76]

J. R. Schrieffer and P. A. Wolff, Relation between the Anderson and Kondo Hamiltonians, Phys. Rev.149, 491 (1966)

1966
[77]

Mæland,Many-body effects and topology in magnets and superconductors, Ph.D

K. Mæland,Many-body effects and topology in magnets and superconductors, Ph.D. thesis, NTNU, Norway (2024)

2024
[78]

Sigrist and K

M. Sigrist and K. Ueda, Phenomenological theory of unconven- tional superconductivity, Rev. Mod. Phys.63, 239 (1991)

1991
[79]

A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B78, 195125 (2008)

2008
[80]

Read and D

N. Read and D. Green, Paired states of fermions in two di- mensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect, Phys. Rev. B61, 10267 (2000)

2000
[81]

P. B. Allen and R. C. Dynes, Transition temperature of strong- coupled superconductors reanalyzed, Phys. Rev. B12, 905 (1975)

1975
[82]

Brekke, A

B. Brekke, A. Brataas, and A. Sudbø, Two-dimensional alter- magnets: Superconductivity in a minimal microscopic model, Phys. Rev. B108, 224421 (2023)

2023
[83]

Mæland, B

K. Mæland, B. Brekke, and A. Sudbø, Many-body effects on superconductivity mediated by double-magnon processes in al- termagnets, Phys. Rev. B109, 134515 (2024)

2024
[84]

Parente, A

V . Parente, A. Tagliacozzo, F. von Oppen, and F. Guinea, Electron-phonon interaction on the surface of a three- dimensional topological insulator, Phys. Rev. B88, 075432 (2013)

2013
[85]

E. V . Gorbar, V . A. Miransky, I. A. Shovkovy, and P. O. Sukha- chov, Dirac semimetalsA 3Bi(A=Na,K,Rb)asZ 2 Weyl semimetals, Phys. Rev. B91, 121101(R) (2015)

2015
[86]

E. V . Gorbar, V . A. Miransky, I. A. Shovkovy, and P. O. Sukha- chov, Surface Fermi arcs inZ 2 Weyl semimetalsA 3Bi(A= Na, K, Rb), Phys. Rev. B91, 235138 (2015)

2015
[87]

S.-Y . Xu, C. Liu, S. K. Kushwaha, R. Sankar, J. W. Krizan, I. Belopolski, M. Neupane, G. Bian, N. Alidoust, T.-R. Chang, H.-T. Jeng, C.-Y . Huang, W.-F. Tsai, H. Lin, P. P. Shibayev, F.- C. Chou, R. J. Cava, and M. Z. Hasan, Observation of Fermi arc surface states in a topolo...

2014
[88]

Kargarian, M

M. Kargarian, M. Randeria, and Y .-M. Lu, Are the surface Fermi arcs in Dirac semimetals topologically protected?, Proc. Natl. Acad. Sci. U.S.A.113, 8648 (2016)

2016
[89]

Changdar, O

S. Changdar, O. Suvorov, A. Kuibarov, S. Thirupathaiah, G. Shipunov, S. Aswartham, S. Wurmehl, I. Kovalchuk, K. Koepernik, C. Timm, B. B¨uchner, I. C. Fulga, S. Borisenko, and J. v. d. Brink, Topological nodali-wave superconductivity in PtBi2, arXiv:2507.01774 (2025)

2025
[90]

Kvorning, T

T. Kvorning, T. H. Hansson, A. Quelle, and C. M. Smith, Pro- posed Spontaneous Generation of Magnetic Fields by Curved Layers of a Chiral Superconductor, Phys. Rev. Lett.120, 217002 (2018)

2018
[91]

P. O. Sukhachov, F. von Oppen, and L. I. Glazman, Andreev Reflection in Scanning Tunneling Spectroscopy of Unconven- tional Superconductors, Phys. Rev. Lett.130, 216002 (2023)

2023
[92]

P. O. Sukhachov, F. von Oppen, and L. I. Glazman, Tunneling spectra of impurity states in unconventional superconductors, Phys. Rev. B108, 024505 (2023)

2023
[93]

Panigrahi, V

A. Panigrahi, V . Poliakov, and L. Levitov, Tunneling density of states in exotic superconductors and spatial patterns of particle- hole interference, arXiv:2503.16168 (2025)

2025 arXiv
[94]

Enzner, J

S. Enzner, J. Berges, A. Schobert, D. Oh, M. Kang, R. Comin, R. Thomale, T. O. Wehling, D. Di Sante, and G. Sangio- vanni, Phonon fluctuation diagnostics: Origin of charge order in A V3Sb5 kagome metals, arXiv:2504.07883 (2025)

2025 arXiv
[95]

Okugawa and S

R. Okugawa and S. Murakami, Dispersion of Fermi arcs in Weyl semimetals and their evolutions to Dirac cones, Phys. Rev. B89, 235315 (2014)

2014
[96]

N. H. Aase, K. Mæland, and A. Sudbø, Multiband strong- coupling superconductors with spontaneously broken time- reversal symmetry, Phys. Rev. B108, 214508 (2023)

2023
[97]

O’Leary, Z

E. O’Leary, Z. Li, L.-L. Wang, B. Schrunk, A. Eaton, P. C. Can- field, and A. Kaminski, Topography of Fermi Arcs in t-PtBi 2 Using High Resolution Angle-resolved Photoemission Spec- troscopy, arXiv:2503.08841 (2025)

2025 arXiv
[98]

Kuibarov, S

A. Kuibarov, S. Changdar, A. Fedorov, R. Lou, O. Su- vorov, V . Misheneva, L. Harnagea, I. Kovalchuk, S. Wurmehl, B. B¨uchner, and S. Borisenko, Measuring superconducting arcs by ARPES, arXiv:2505.09347 (2025)

2025 arXiv
[99]

E. W. Hodt, P. Sukhachov, and J. Linder, Interface-induced magnetization in altermagnets and antiferromagnets, Phys. Rev. B110, 054446 (2024)

2024

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [60]

Mæland, S

K. Mæland, S. Abnar, J. Benestad, and A. Sudbø, Topologi- cal superconductivity mediated by magnons of helical magnetic states, Phys. Rev. B108, 224515 (2023)

2023

[2] [1]

There aredrphonon modes, wheredis the dimensionality andris the number of atoms in the basis

General framework Phonon dispersions can be derived in specific lattice sys- tems using the system’s symmetries through a force constant approach [57–61]. There aredrphonon modes, wheredis the dimensionality andris the number of atoms in the basis. dof the phonon modes are acoustic, with zero energy at zero momentum, while the remainingd(r−1)modes are opt...

[3] [2]

cos(ky) + 2 cos √ 3kx 2 ! cos ky 2 # ,(A6) gk =t o

Those parameters giveλ≈0.38andT c ≈19.7K ift= 1eV . number inversely proportional to the standard deviation of the atomic orbitals [60, 61]. Appendix C provides more details of the derivation of the EPC. V . SUPERCONDUCTIVITY A Schrieffer-Wolff transformation [64, 65] yields an effec- tive electron-electron interaction mediated by the phonons. Focusing on...

[4] [3]

There are three mirror planes meeting at120 ◦ normal to thexyplane, with the inter- section being thezaxis

Hexagonal lattice In our case, we consider a 3D crystal with a one-atomic ba- sis (r= 1andα, βindices become superfluous) with symme- tries given by the P31m space group. There are three mirror planes meeting at120 ◦ normal to thexyplane, with the inter- section being thezaxis. Also, there is a three-fold rotational symmetry about thezaxis. For rotations ...

[5] [4]

[62, 63] for other studies of phonons with open boundary conditions, which includes examples of the range of phonon energies for certain materials

Phonon spectrum with one open boundary condition See Refs. [62, 63] for other studies of phonons with open boundary conditions, which includes examples of the range of phonon energies for certain materials. Reference [63] also discusses surface phonons and enhanced EPC at the surface. The origin is that interatomic distances may change close to the surfac...

[6] [5]

The combinationsg m kk′gm −k,−k′ and gm k,−k′gm −k,k′ both contain productse ziµ qmez′ iν −q,m

Gauge dependence Since it is ¯V FS kk′ =− P m gm kk′gm −k,−k′/ωk−k′,m +P m gm k,−k′gm −k,k′/ωk+k′,m which enters in the gap equation we focus on its behavior. The combinationsg m kk′gm −k,−k′ and gm k,−k′gm −k,k′ both contain productse ziµ qmez′ iν −q,m. The phonon eigenvectors have the propertyˆe −q,m = [ˆeqm]∗ which can be shown from the fact that the d...

[7] [6]

Hence, they correspond to moving all ions the same amount inx, y, orzdirection

Zero momentum transfer The three phonon modes with zero energy atq= 0in the slab geometry have eigenvectors with elements1/Lat the x, y, orzposition at each layer and otherwise zero. Hence, they correspond to moving all ions the same amount inx, y, orzdirection. So, both for the in-plane and out-of-plane type EPC, the numerator inV FS kk′ goes to zero whe...

[8] [7]

In both cases, we see that the cou- pling is negligible fork ′ in the bulk and top surface parts of the FS, supporting the claim in Sec

Bulk and surface separability and origin of gap symmetry Figures 4(d)-4(g) show ¯V FS kk′ as a function ofk ′ on the full FS withkfixed at the center of the Fermi arc on bottom sur- face and halfway between the center and endpoint of the bot- tom surface Fermi arc. In both cases, we see that the cou- pling is negligible fork ′ in the bulk and top surface ...