Nodal loop and nodal surface states in the Ti3Al family of materials

Topological metals and semimetals are new states of matter which are attracting great interest in current research. Here, based on first-principles calculations and symmetry analysis, we propose that the family of titanium-based compounds ${\mathrm{Ti}}_{3}X$ ($X$=Al, Ga, Sn, Pb) are unexplored topological metals. These materials feature the coexistence of a nodal loop and a nodal surface in their low-energy band structures. Taking ${\mathrm{Ti}}_3\mathrm{Al}$ as an example, we show that the material has an almost ideal nodal loop in the sense that the loop is close to the Fermi level and it is nearly flat in energy with energy variation $<0.25$ meV. The loop is protected by one of the two independent symmetries: the combined space-time-inversion symmetry and the mirror-reflection symmetry. The nodal surface at the $k_z=\pi$ plane is guaranteed by the nonsymmorphic screw rotational symmetry and the time-reversal symmetry. We discuss the effect of spin-orbit coupling and construct an effective model for describing the nodal loop. Our findings indicate that the ${\mathrm{Ti}}_3\mathrm{Al}$ family of compounds can serve as an excellent material platform for studying new topological phases and, particularly, the interplay between nodal loop and nodal surface fermions.