Hourglass Dirac chains enable intrinsic topological superconductivity in nonsymmorphic silicides

Abstract

Nonsymmorphic crystalline symmetries provide a robust route to symmetry-protected electronic topology, yet their role in stabilizing intrinsic topological superconductivity remains largely unexplored. Here, we report \ch{TaPtSi} as a new member of the superconducting nonsymmorphic silicide family, characterized via AC transport, magnetization, heat capacity, and muon spin rotation/relaxation ($μ$SR) measurements. Zero field $μ$SR reveals spontaneous internal magnetic fields below $T_{\rm c}$, establishing time reversal symmetry breaking in \ch{TaPtSi}. First principles calculations on \ch{TaPtSi} and its isostructural nonsymmorphic superconducting analogues reveal the presence of symmetry-protected hourglass dispersions. The "necks" of these dispersions form Dirac nodal rings and chains that reside near or intersect the Fermi level. Guided by Ginzburg Landau symmetry analysis, we identify an internally antisymmetric non unitary triplet pairing state as the unique ground state consistent with the experimental phenomenology. Based on Bogoliubov de Gennes calculations, we further demonstrate that this state supports Majorana surface modes, establishing its intrinsically topological nature. These results reveal a systematic route by which nonsymmorphic symmetry drives the interplay between hourglass Dirac chain topology and unconventional triplet pairing, positioning equiatomic silicides as a unified materials platform for intrinsic topological superconductivity.

Figures (11)

• Figure 1: Structural and superconducting properties of TaPtSi: (a) The crystal structure of $MT$Si ($M$ = Ta, Nb; $T$ = Pt, Os). (b) Rietveld refined powder XRD pattern for TaPtSi, where observed data, fitted line, their difference, and the Bragg positions are mentioned. (c) The plot of the low-temperature AC resistivity data shows a sharp zero drop in resistivity. (d) Magnetic moment versus temperature measured in ZFCW and FCC modes, showing superconductivity at 3.64(1) K. The inset displays the M-H loop at 1.8 K. (e) Variation of upper critical field with reduced temperature ($T/T_{\rm c}$), fitted with the GL equation. (f) The temperature-dependent electronic specific heat in the low-temperature region, well-fitted with the isotropic s-wave model. Inset: The data in the normal region is well described by the Debye model.
• Figure 2: Time-reversal symmetry breaking with fully gapped superconductivity from $\mu$SR in TaPtSi: (a) The zero-field muon spectra recorded at 0.26 K and 5 K. Solid lines exhibit the fitting of the asymmetry spectra using Eq. \ref{['Eq:kubo']}. (b) The temperature variations of $\Lambda$ and $\Delta$ (Inset) along with the error bar, showing an increase in $\Lambda$ below $T_{\rm c}$, $\Delta$ being constant. (c) Temperature variation of the TF muon asymmetry spectra below $T_{\rm c}$, and above $T_{\rm c}$, under 20 mT magnetic fields. (d) The Brandt fitting yielded the fitted parameters $\lambda^{-2}$, which varied with temperature. The orange line denotes the optimal fit of the data with Eq. \ref{['Eq:swavemuon']}.
• Figure 3: Electronic band structure of the silicides $MT$Si ($M$ = Ta, Nb; $T$ = Pt, Os): (a) Three-dimensional bulk Brillouin zone (BZ) and the (100) surface BZ of the primitive orthorhombic lattice, with high-symmetry points and paths indicated by red dots and blue lines, respectively. (b,c,d) Bulk electronic band structures of TaPtSi, TaOsSi, and NbOsSi obtained with spin–orbit coupling (SOC), exhibiting hourglass-type dispersions along the $S-X$ and $S-R$ directions of the BZ. (e,f,g) Fermi surfaces of TaPtSi, TaOsSi, and NbOsSi calculated including SOC, illustrating the multiple sheets spanning the BZ.
• Figure 4: Hourglass dispersions, Dirac rings and surface states of $MT$Si ($M$ = Ta, Nb; $T$ = Pt, Os): (a,c,e) Hourglass-type band dispersions of TaPtSi, TaOsSi, and NbOsSi along the high-symmetry path $S-R$. (b,d,f) Corresponding hourglass dispersions along the $S-K$ direction, where $K$ is the midpoint between $T$ and $R$ (see inset of panel (f)). Type-I and type-II Dirac crossings are highlighted by red and orange markers, respectively. The inset in (b) shows distribution of the Dirac ring (white) for TaPtSi, encircling the $S$ point. The color map indicates the magnitude of the local band gap in meV. The inset in (f) illustrates the fourfold-degenerate Dirac ring generated by the neck points (red dots) of the hybrid hourglass band structure on the $k_x=\pi$ plane. Dirac points pinned at the Fermi energy are highlighted by black dotted circles in (d) and (f). (g,h,i) Surface-state spectra along high-symmetry paths of the projected (100) surface Brillouin zone, highlighting the nontrivial drumhead surface states emerging around the $\bar{Y}$ point. (j,k,l) Constant-energy contours of the surface spectral weight for TaPtSi at –0.10 eV, TaOsSi at –0.01 eV, and NbOsSi at 0.0 eV, illustrating the momentum-space extent and morphology of the drumhead states.
• Figure 5: Topological superconductivity enabled by hourglass topology: (a) Normal-state band dispersion exhibiting the hourglass feature along the $k_y$ direction for a minimal model capturing Dirac hourglass connectivity. (b) Surface spectral function of the Dirac hourglass semimetal represented by the minimal model, with the color scale representing the surface spectral weight, A$_N(\bar{k},\omega)$. (c) Bogoliubov–de Gennes quasiparticle spectrum of the internally antisymmetric nonunitary triplet (INT) superconducting state in the hourglass model, showing a fully gapped superconducting state. (d) Surface spectral function of the hourglass semimetal in the INT superconducting phase, A$_S(\bar{k},\omega)$, revealing linearly dispersing Majorana surface modes. Here, $\omega$ denotes the surface energy, and $\bar{k}$ represents the momentum along the $\hat{z}$ direction of the surface Brillouin zone. The model parameters used are $t_a=0.5t$, $t_c=0.5t$, $\lambda=0.8t$, $t_\perp=0.7t$, $\lambda_\perp=0.4t$, $m_1=0.6t$, $m_2=0.8t$, $\lambda_{\perp}^\prime=0.3t$, and $\mu=3.0t$. For the INT pairing, the triplet vector components are chosen as $\eta_x=\tfrac{1}{\sqrt{2}}$, $\eta_y=\tfrac{e^{i\pi/4}}{\sqrt{2}}$, and $\eta_z=0$.