{"id":"bf85cdba-2549-4982-91af-aab73d4e20e3","arxiv_id":"2502.07475","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"TiIrGe and HfIrGe are fully gapped, time-reversal-preserving superconductors whose nonsymmorphic crystal symmetry is predicted to produce hourglass Dirac chain topology and helical surface states.","lead":"The paper reports that TiIrGe and HfIrGe are conventional superconductors with full gaps and preserved time-reversal symmetry below about 2.2 K and 5.6 K, and uses density functional theory to predict hourglass-shaped band crossings and helical topological surface states protected by their crystal symmetry.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Z2=1 on an unproven gapless ky=0 plane and PBE-level Fermi energy carry the 'ideal platform' claim; the invariant and surface-state crossing are asserted, not demonstrated.","rationale":"The reader's conditional verdict already identifies the same load-bearing weakness: the Z2=1 invariant is defined on a single momentum plane in a material with no global gap, and the supporting calculation is not shown. My independent reading confirms that this is the most consequential unresolved point. The bulk superconducting characterization is coherent and strongly supports conventional, fully gapped, time-reversal-preserving superconductivity; the hourglass symmetry analysis is plausible and follows the established nonsymmorphic logic. The vulnerability is specifically the topological surface-state claim: the paper states that the ky=0 plane is gapped and Z2=1, but supplies neither the plane gap nor the invariant calculation, and the only shown Fermi-level-crossing surface states come from a DFT-PBE calculation with no sensitivity check. Since the abstract and conclusion build the 'prime candidates for topological superconductivity' claim on exactly these surface states crossing the Fermi level, the conditional recommendation is appropriate: the Z2 computation and gap on the ky=0 plane should be supplied, or the claims should be tempered. The concrete test proposed here would settle whether the concern lands by checking both the invariant and the robustness of the Fermi-level crossing to typical PBE errors.","tokens_in":23098,"tokens_out":6277,"duration_ms":61778,"concrete_test":"Recompute the Z2 invariant on the ky=0 plane from the published Wannier tight-binding model using a Wilson loop, and explicitly plot the local gap over that entire plane to confirm it is fully gapped. Then rerun the (010) slab spectral function with a rigid Fermi-level shift of +/-0.1 eV, and ideally with a hybrid functional, to check whether a helical surface state still crosses the Fermi level. If the ky=0 plane is gapless, or if the surface-state crossing disappears under the shift, the 'ideal platform' conclusion should be downgraded to a candidate whose topological surface states have not been shown to be at the Fermi level.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the inference from DFT to a nontrivial Z2=1 and to helical surface states dispersing across the Fermi level. The authors concede that the compounds have no global gap (Sec. 2, 'Topology of electronic band structure') and define the invariant on the ky=0 plane (SM, 'Band structure and topology'), but they do not show that this plane is fully gapped, nor do they show a Wilson loop or parity eigenvalues for the four TRIM points in that plane. A plane-restricted invariant in a metal is only well-defined if that plane is insulating; if the ky=0 plane is gapless, or if PBE places the Fermi level incorrectly, the Z2=1 claim and the resulting 'ideal platform' conclusion do not follow. The surface spectrum in Fig. S5 is presented as evidence, but no independent functional or rigid-shift check is given, and the drumhead states in Fig. 3f/i are shown at -0.138 eV / -0.130 eV rather than at EF. This is the single point on which the 'prime candidates for topological superconductivity' claim rests.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript reports a combined experimental and first-principles study of the ternary germanides TiIrGe and HfIrGe (orthorhombic TiNiSi-type, space group Pnma). Resistivity, magnetization, and specific heat measurements establish bulk type-II superconductivity with Tc = 2.24(5) K (TiIrGe) and 5.64(4) K (HfIrGe); transverse-field muSR is consistent with a fully gapped s-wave response with penetration depths of about 273 and 246 nm, and zero-field muSR finds no spontaneous magnetic field, ruling out time-reversal-symmetry breaking in the superconducting state. DFT (PBE) calculations predict glide-mirror-protected hourglass dispersions along S-X, S-R, and S-K with a Dirac ring around the S point in the kx = pi plane, drumhead surface states, and a claimed Z2 = 1 invariant on the ky = 0 plane that yields helical surface states crossing the Fermi level. The authors conclude that the coexistence of conventional bulk s-wave superconductivity with these topological surface states makes MIrGe an ideal stoichiometric platform for proximity-induced topological superconductivity. The paper explicitly acknowledges the absence of a global bulk gap and the reliance on the plane-restricted Z2 definition.","tokens_in":23343,"tokens_out":16388,"duration_ms":127751,"significance":"If the topological predictions hold, the paper identifies a rare materials class in which a conventional, fully gapped bulk superconductor coexists with nonsymmorphic-protected hourglass Dirac-chain topology and helical spin-textured surface states, making these compounds attractive stoichiometric candidates for proximity-based topological superconductivity. The experimental half is a genuine strength: bulk superconductivity is established by four independent probes, the muSR analysis follows standard protocols, the ZF-muSR measurement addresses TRS breaking directly, and the McMillan-based Tc estimate (2.71 K and 5.08 K with mu* = 0.10) agrees closely with experiment without tuning. The hourglass symmetry argument is explicit and checkable. The main evidentiary weakness is that the topological half is purely predictive: no ARPES or STM data test the surface states, so the weight of the central claim rests entirely on the completeness of the DFT analysis, which is presently incomplete in the specific ways detailed below.","major_comments":[{"comment":"The Z2 = 1 claim for the ky = 0 plane is load-bearing for the paper's central conclusion, but it is asserted rather than demonstrated. The manuscript states that MIrGe has no global gap with SOC and that Z2 is 'well-defined on the ky = 0 plane,' yet it does not show the band structure restricted to that plane, does not establish that the plane is gapped at the Fermi energy, and provides no Wilson loop or parity eigenvalues at the four time-reversal-invariant momenta in that plane. The only surface-state evidence offered is Fig. S5a for HfIrGe, with no equivalent spectrum for TiIrGe. A plane-restricted Z2 index is only meaningful if that plane is insulating, so this missing check is not cosmetic; without it, the helical-surface-state claim and the 'prime candidates for topological superconductivity' conclusion do not follow.","section":"Sec. 2, Topology of electronic band structure; SM, Band structure and topology"},{"comment":"The position of the predicted surface states relative to the Fermi level is not robustly established. The drumhead states are reported at -0.138 eV (TiIrGe) and -0.130 eV (HfIrGe), and the helical-state constant-energy contour is shown at -0.090 eV (Fig. S5b), yet the abstract and conclusions assert that the helical surface states 'disperse across the Fermi level.' These energies lie within the typical tens-of-meV accuracy of PBE band fillings; no rigid chemical-potential shift, alternative exchange-correlation functional, or Hubbard-U check is provided, and no experimental ARPES comparison exists. The authors should either supply such a sensitivity analysis or qualify the 'ideal platform' claim proportionately.","section":"Sec. 2, Fig. 3 and SM Fig. S5"},{"comment":"The main text and the SM disagree on the reliability of the TiIrGe specific-heat gap fit. The main text says the electronic specific heat is 'well fitted with the fully gapped weak-coupling BCS model' and quotes Delta/kBTc = 1.47(2), whereas the SM states that for TiIrGe 'the fitting ... is not determined accurately due to insufficient data points at low temperatures.' The quantitative agreement between methods is also loose: the specific-heat ratios are 1.47(2) and 2.04(2), while the muSR ratios are 1.66(7) and 1.68(2), and the discrepancies are not discussed. The s-wave full-gap conclusion is probably correct, but the main text should carry the SM caveat and the spread of gap values should be reconciled or explicitly discussed.","section":"Sec. 2, Muon spin rotation and relaxation; SM, Specific heat"}],"minor_comments":[{"comment":"The printed McMillan formula, Tc = omega_log 1.2 exp[...], is missing the division sign; the standard Allen-Dynes form is Tc = (omega_log/1.2) exp[...].","section":"Sec. 3, Discussions"},{"comment":"The sentence describing SOC-induced splittings cites Figure 1d,e, but those panels are the without-SOC band structures; the with-SOC panels are Figure 1h,i.","section":"Sec. 2, Electronic band structure"},{"comment":"For HfIrGe the quoted Delta(0) = 0.75(2) meV together with Tc = 5.64(4) K implies Delta(0)/kBTc ~ 1.54, not the reported 1.68(2); this internal inconsistency should be corrected.","section":"Sec. 2, Muon spin rotation and relaxation"},{"comment":"The two reported upper critical fields for HfIrGe, 1.36(1) T from magnetization and 2.04(1) T from resistivity, differ by roughly 50% and are presented without comment.","section":"Sec. 2, Critical fields"},{"comment":"The label 'Surface Fermi arcs' is imprecise for drumhead surface states of a Dirac ring in a time-reversal-invariant system; suggest 'constant-energy contours of the drumhead surface states.'","section":"Fig. 3 caption"},{"comment":"The stated detection limit of 'up to 1 microT' for spontaneous internal fields is not derived; please state how this bound follows from the zero-field depolarization data.","section":"Sec. 2, ZF-muSR"},{"comment":"The surface-state predictions pertain to specific surfaces, but the measured samples are polycrystalline; a sentence clarifying that single crystals will be required to test the (100)/(010) surface states with ARPES or STM would be helpful.","section":"Sec. 4, Summary and conclusion"}],"recommendation":"major_revision","confidential_remarks":"The experimental superconductivity characterization is the strongest part of the paper and is essentially publishable as is. The revision burden is entirely on the topological half: the plane-restricted Z2 invariant must be backed by an explicit demonstration that the ky = 0 plane is gapped (plus the invariant computation and the TiIrGe surface spectrum), or the conclusions need to be scaled back. There is also a modest internal-consistency problem in the quoted gap values that the authors should clean up. If the Z2 evidence cannot be produced, I would still see the paper as publishable with the 'ideal platform' language softened to a conditional proposal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a careful experimental characterization of two stoichiometric superconductors, TiIrGe and HfIrGe, with a DFT-based prediction of hourglass Dirac-chain topology. The muSR, specific heat, resistivity, and magnetization work is solid and internally consistent: both compounds come out as conventional type-II, fully gapped s-wave superconductors with preserved TRS. That alone is a useful data point, and the penetration depth and Uemura analysis are in line with similar systems like (Zr,Hf)IrSi.\n\nWhat's new is the combination of that bulk characterization with a first-principles claim that these are \"prime candidates for topological superconductivity.\" The hourglass dispersion and Dirac chain argument is plausible, and the symmetry analysis along S-R is well explained. But the load-bearing step—the Z2=1 invariant on the ky=0 plane—is asserted in one sentence and never demonstrated. No Wilson loop, no parity eigenvalues, no check that the ky=0 plane is actually gapped. Given the compounds have no global gap, that plane-restricted invariant is only meaningful if that plane is insulating, and the paper doesn't show it. The drumhead states in Fig. 3 are at -0.138/-0.130 eV, not at EF, and the claim that helical surface states cross the Fermi level relies on Fig. S5, which again lacks the supporting invariant calculation. The PBE Fermi level position is also not checked against a hybrid functional or rigid shift.\n\nThe specific heat gap for TiIrGe is admittedly unreliable due to too few low-temperature points; that's a minor issue because the muSR gap is fine.\n\nThe abstract and conclusion overstate the result. This is a candidate platform, not a demonstrated topological superconductor. The 1987 superconductivity paper should be acknowledged directly in the abstract.\n\nRecommendation: major revision. Ask the authors to show the Z2 calculation (Wilson loop or parity eigenvalues) and the gap on ky=0, or temper the title and abstract to \"candidate.\" The experimental part deserves publication; the topological claim needs to be either proven or softened.\n\nWho benefits: people working on stoichiometric topological superconductor candidates, muSR practitioners. It deserves a serious referee.","headline":"Solid muSR/thermodynamic study of two superconductors, but the 'ideal topological platform' claim rests on an unshown Z2 calculation; needs either the invariant or tempered claims.","tokens_in":23899,"tokens_out":2614,"would_cite":true,"duration_ms":23703,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"TiIrGe and HfIrGe combine bulk s-wave superconductivity with nonsymmorphic hourglass topology and helical Fermi-level surface states, making them prime candidates for topological superconductivity.","keywords":["topological superconductivity","hourglass Dirac chain","nonsymmorphic glide mirror","TiIrGe","HfIrGe","muon spin rotation","s-wave superconductor","Z2 invariant"],"falsifier":"Angle-resolved photoemission on the (010) surface of a single crystal that finds no spin-textured surface state crossing the Fermi level at the predicted energy, or a Wilson-loop calculation on the ky = 0 plane that yields Z2 = 0 instead of 1, would falsify the central claim.","tokens_in":22893,"feed_emoji":"🧲","tokens_out":8329,"duration_ms":74893,"temperature":0.7,"pith_summary":"This paper argues that the ternary germanides TiIrGe and HfIrGe combine two properties that are rarely found in one stoichiometric crystal: conventional bulk superconductivity and symmetry-protected topological surface states at the Fermi level. Muon-spin rotation, magnetization, resistivity, and specific heat measurements show fully gapped, time-reversal-preserving type-II s-wave superconductivity with transition temperatures of 2.24(5) K and 5.64(4) K. First-principles band-structure calculations predict hourglass-shaped bulk dispersions whose necks form a Dirac chain, a ring of fourfold-degenerate Dirac points protected by a nonsymmorphic glide mirror, together with a nontrivial Z2 invariant that produces helical Dirac surface states crossing the Fermi level. If these calculations are right, the bulk condensate can open a superconducting gap on the topological surface states directly, without doping, interfaces, or heterostructure engineering.","feed_headline":"Bulk s-wave pairing meets hourglass topology in TiIrGe and HfIrGe","feed_subtitle":"The compounds' helical surface states could host topological superconductivity without doping or interface engineering.","key_machinery":"The load-bearing object is the nonsymmorphic glide mirror symmetry Gx: (x, y, z) -> (-x + 1/2, y + 1/2, z + 1/2) and its enforced eigenvalue redistribution along the S-R path. Because $Gx^{2}$ = $e^{{-ikz}}$ on that path, the glide eigenvalues change from +-1 at S to +-i at R, forcing a band crossing that appears as an hourglass dispersion; the neck points of these hourglasses close into a Dirac ring, the Dirac chain, around S in the kx = pi plane. A second ingredient is the Z2 = 1 invariant computed on the ky = 0 plane, the only plane where a gap exists, which produces helical Dirac surface states. The argument combines symmetry analysis with tight-binding interpolation of first-principles bands, and matches the gap symmetry from specific heat and muon-spin rotation data.","core_discovery":"The central claim is that MIrGe (M = Ti, Hf) are prime candidates for topological superconductivity: they are conventional, fully gapped, weak-coupling type-II superconductors with Tc = 2.24(5) K for TiIrGe and 5.64(4) K for HfIrGe, isotropic s-wave gaps, and no time-reversal symmetry breaking. In the normal state, DFT calculations including spin-orbit coupling reveal nonsymmorphic glide-mirror-protected hourglass dispersions whose necks form a continuous Dirac chain, a ring of fourfold-degenerate Dirac points in the kx = pi plane, which generates drumhead-like surface states. A Z2 = 1 invariant on the ky = 0 plane yields isolated Dirac surface states with helical spin texture on the (010) surface, well separated from bulk states and crossing the Fermi level. The paper concludes that the coexistence of the bulk s-wave gap with these helical topological surface states makes the compounds a stoichiometric platform for proximity-induced topological superconductivity and potentially Majorana physics, with the topological features verifiable by ARPES or STM/STS.","pith_inferences":["Beyond the paper: a Wilson-loop or parity-eigenvalue calculation on the ky = 0 plane, which the paper does not show, would settle whether the Z2 = 1 assignment is genuine; the text currently states the index without displaying the calculation.","Beyond the paper: if the predicted surface gap appears, planar Josephson junctions or point-contact spectroscopy on these compounds could look for zero-bias conductance peaks or fractional Josephson signatures as the decisive Majorana evidence, a step the paper mentions only as a long-term possibility.","Beyond the paper: the same nonsymmorphic symmetry data suggest that isostructural sister compounds, such as ZrIrGe, would also host hourglass Dirac chains, turning MIrGe into a family-level platform rather than a two-material exception.","Beyond the paper: the muon-derived penetration-depth discrepancy for HfIrGe hints that vortex-state and Meissner-state measurements may be probing different effective parameters; a small-angle neutron scattering or tunnel-diode resonator measurement could resolve the origin."],"forward_implications":["ARPES and STM/STS should observe the predicted hourglass dispersions along S-X, S-R, and S-K, together with drumhead surface states where the Dirac chain projects onto the (100) surface.","The helical Z2 surface states crossing the Fermi level should acquire a proximity-induced superconducting gap from the bulk s-wave condensate, appearing as a distinct surface gap in Andreev reflection or tunneling spectroscopy.","Because the topological surface states are intrinsic to the stoichiometric crystal, the platform avoids the doping- and interface-related fragility of earlier Bi2Se3- and SnTe-based proposals.","The phonon calculation gives Tc values of 2.71 K for TiIrGe and 5.08 K for HfIrGe, close to the measured values, supporting the conventional electron-phonon pairing picture used in the argument.","Ginzburg-Landau symmetry analysis says all superconducting order parameters except the fully symmetric A1 s-wave channel have nodes, so a signature of nodal pairing would point away from the conventional s-wave scenario established here."],"supporting_citations":[{"why":"Supplies the hourglass-fermion concept and the nonsymmorphic-symmetry protection argument the paper applies to MIrGe.","marker":"[5]"},{"why":"Gives the hourglass Dirac-loop and drumhead surface-state mechanism and the proximity-to-topological-superconductivity idea the paper leans on.","marker":"[10]"},{"why":"Is the comparative muon-spin study of Dirac-semimetallic silicides whose s-wave and time-reversal-symmetry analysis method is applied here.","marker":"[33]"},{"why":"Reports the crystal structure and superconductivity of the MTGe germanide family, the materials under study.","marker":"[38]"},{"why":"Is the density-functional exchange-correlation functional used for all band-structure and topology calculations.","marker":"[40]"},{"why":"Underpins the muon-spin rotation and relaxation analysis that establishes the full gap and preserved time-reversal symmetry.","marker":"[42]"},{"why":"Provides the modified McMillan formula used to compute Tc from phonon spectra, supporting conventional pairing.","marker":"[52]"},{"why":"Demonstrates a bulk-condensate-induced surface superconducting gap in a related topological compound, the experimental template for the platform claim.","marker":"[56]"}],"fun_headline_variants":["Hourglass Dirac chains make TiIrGe and HfIrGe topological superconductor candidates","s-wave pairing meets helical surface states in MIrGe superconductors","Stoichiometric MIrGe: hourglass Dirac chain and topological surface states","TiIrGe and HfIrGe: hourglass Dirac ring and s-wave gap suggest Majorana physics","Nonsymmorphic symmetry in MIrGe protects hourglass Dirac chains and helical states"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The key unproven step is that the ky = 0 plane of the Brillouin zone is fully gapped with a nontrivial index and that the calculated Fermi energy places the helical surface states exactly at the Fermi level; the paper asserts both but shows neither the plane's gap nor the invariant's explicit calculation.","fun_headline_variants_meta":{"raw":{"variants":["Hourglass Dirac chains make TiIrGe and HfIrGe topological superconductor candidates","s-wave pairing meets helical surface states in MIrGe superconductors","Stoichiometric MIrGe: hourglass Dirac chain and topological surface states","TiIrGe and HfIrGe: hourglass Dirac ring and s-wave gap suggest Majorana physics","Nonsymmorphic symmetry in MIrGe protects hourglass Dirac chains and helical states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001145,"raw_usage":{"total_tokens":4808,"prompt_tokens":1058,"completion_tokens":3750,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":674,"completion_tokens_details":{"reasoning_tokens":3638}},"tokens_in":674,"tokens_out":3750,"duration_ms":25068,"temperature":1.0,"reasoning_tokens":3638,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T12:36:31.480285+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Angle-resolved photoemission on the (010) surface of a single crystal that finds no spin-textured surface state crossing the Fermi level at the predicted energy, or a Wilson-loop calculation on the ky = 0 plane that yields Z2 = 0 instead of 1, would falsify the central claim.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the hourglass Dirac-loop and drumhead surface-state mechanism and the proximity-to-topological-superconductivity idea the paper leans on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Is the comparative muon-spin study of Dirac-semimetallic silicides whose s-wave and time-reversal-symmetry analysis method is applied here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Underpins the muon-spin rotation and relaxation analysis that establishes the full gap and preserved time-reversal symmetry."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Demonstrates a bulk-condensate-induced surface superconducting gap in a related topological compound, the experimental template for the platform claim."}],"review_version":1}