Fig. 1: Design of ideal Kramers nodal lines in intercalated transition metal dichalcogenides and their material properties.

a Illustration of the indium intercalation into the inversion symmetric 2H-TaS₂ (with one inversion center denoted by the empty circle). b Crystal structures of In_xTaS₂ (left: x = 1/2, right: x = 1). Both exhibit broken inversion symmetry. The transparent magenta horizontal planes denote the M_z mirror symmetry that is crucial to the formation of the Kramers nodal lines (KNLs). c Three-dimensional (3D) and two-dimensional (2D) projected surface Brillouin zone of the crystal structure in (b), where the purple lines indicate the KNL directions (M–Γ–A–L). d Low-energy dispersions of the spin-orbit coupling splitting (red and blue) and Kramers nodal lines (green tubes) of the point group of the intercalated transition metal dichalcogenide family, D_3h (\({{{{\rm{G}}}}}_{24}^{11}\): R₇, R₈)⁶. S_↑ and S_↓ represent pseudospin up and down. The Hamiltonian takes the form of \(i{\alpha }_{1}({k}_{+}^{2}-{k}_{-}^{2}){k}_{z}{\sigma }_{x}-{\alpha }_{1}({k}_{+}^{2}+{k}_{-}^{2}){k}_{z}{\sigma }_{y}+i{\alpha }_{2}({k}_{+}^{3}-{k}_{-}^{3}){\sigma }_{z}\), where α₁ = −0.5, α₂ = 0.2, k_z = 0, and k_± = k_x ± ik_y. e Ab initio 3D voxel-style Fermi surface of In_1/2TaS₂ (left) and 2D Bloch spectral function (BSF) calculated at the k_z = 0 and π slices at the Fermi level (right). The 3D red and blue voxels denote high-intensity values of the BSF, which has turned into a Lorentzian-like continuum due to the random occupation of the indium lattice site marked in blue/white in panel (b). The \({\tilde{k}}_{z}\) indicates that the k_z direction is elongated for visualization. f Resistivity as a function of temperature for In_xTaS₂ (blue: x = 1, black: x = 2/3, red: x = 1/2). The inset is a zoom-in view of the low temperature data, showing the superconducting transitions for all three compounds. g The susceptibility as a function of In_xTaS₂ (blue: x = 1, black: x = 2/3). 4πχ approaches the value of −1 at the lowest temperatures, indicating bulk superconductivity in both compounds.