Spin-polarized type-II nodal loop and nodal surface states in hexagonal compounds <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mi>TiO</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math> (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math> = Li, Na, K, Rb)
Tie Yang, Lei Jin, Ying Liu, Xiaoming Zhang, Xiaotian Wang
Abstract
Topological semimetals and metals have been extensively studied in nonmagnetic materials, while those in magnetic materials are still quite rare. Here, based on first-principles calculations, we reveal that a family of hexagonal compounds $X{\mathrm{TiO}}_{2}$ ($X$ = Li, Na, K, Rb) with ferromagnetic ground states host excellent topological electronic structures. In their low-energy band structures, they show two Weyl nodal loops and two Weyl nodal surfaces, where each nodal loop and nodal surface is contributed by the electron states in the same spin channel, namely, spin-polarized nodal loops and nodal surfaces. In particular, the nodal loops in these materials show type-II band dispersions and are robust against spin-orbit coupling because of the protection of the mirror symmetry. These nodal loops are featured with spin-polarized drumhead surface states. We build an effective Hamiltonian, which can well describe the type-II nodal loops in the system. Besides the type-II nodal loops, these materials also possess two spin-polarized nodal surfaces, protected by specific nonsymmorphic symmetry. The findings in this paper can serve as good material platforms to study the fundamental physics of type-II nodal loop and nodal surface fermions in a time-reversal-breaking system.