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.

Nodal-line semimetals in two-dimensional (2D) materials have attracted intense attention recently. From fundamental physics and spintronic application points of view, high Curie temperature ferromagnetic (FM) ones with nodal lines robust against spin-orbit coupling (SOC) are extremely desirable. Here, we propose that the FM ${\mathrm{K}}_2\mathrm{N}$ monolayer is such a Weyl nodal-line semimetal. We show that the ${\mathrm{K}}_2\mathrm{N}$ monolayer is dynamically stable and has a FM ground magnetic state with out-of-plane [001] magnetization. It shows two nodal lines in the low-energy band structures. Both nodal lines are robust against SOC, under the protection of mirror symmetry. We construct an effective Hamiltonian, which can well characterize the nodal lines in the system. Remarkably, the nodal-line semimetal proposed here is distinct from the previously studied ones in that the ${\mathrm{K}}_2\mathrm{N}$ monolayer is a 2D $d^0$-type ferromagnet with the magnetism arising from the partially filled the $p$ orbitals of the N atom, which can bring special advantages in spintronic applications. In addition, the Curie temperature in the ${\mathrm{K}}_2\mathrm{N}$ monolayer is estimated to be 942 K, which is significantly higher than previous FM nodal-line materials. We also find that specific tensile strains can transform the nodal line from type-I to type-II, making its nodal-line characteristics even more interesting.

As a novel type of fermionic state, the hybrid nodal loop with the coexistence of both type-I and type-II band crossings has attracted intense research interest. However, it remains a challenge to realize the hybrid nodal loop in both two-dimensional (2D) materials and in ferromagnetic (FM) materials. Here, we propose a FM hybrid nodal loop in a 2D CrN monolayer. We show that the material has a high Curie temperature ($>600$ K) FM ground state, with the out-of-plane [001] magnetization. It shows a half-metallic band structure with two bands in the spin-up channel crossing each other near the Fermi level. These bands produce both type-I and type-II band crossings, which form a fully spin-polarized hybrid nodal loop. We find the nodal loop is protected by the mirror symmetry and robust against spin-orbit coupling. An effective Hamiltonian characterizing the hybrid nodal loop is established. We further find the configuration of the nodal loop can be shifted under external perturbations such as strain. Most remarkably, we demonstrate that both type-I and type-II Weyl nodes can be realized from such FM hybrid nodal loop by simply shifting the magnetization from out of plane to in plane. Our work provides an excellent candidate to realize a FM hybrid nodal loop and Weyl fermions in 2D material, and is also promising for related topological applications with their intriguing properties.