Multi-loop node line states in ternary MgSrSi-type crystals

Node line band-touchings protected by mirror symmetry (named as m-NLs), the product of inversion and time reversal symmetry S = PT (named as s-NLs), or nonsymmorphic symmetry are nontrivial topological objects of topological semimetals in the Brillouin Zone. In this work, we screened a family of MgSrSi-type crystals using first principles calculations, and discovered that more than 70 members are node-line semimetals. A new type of multi-loop structure was found in AsRhTi that a s-NL touches robustly with a m-NL at some “nexus point”, and in the meanwhile a second m-NL crosses with the s-NL to form a Hopf-link. Unlike the previously proposed Hopf-link formed by two s-NLs or two m-NLs, a Hopf-link formed by a s-NL and a m-NL requires a minimal three-band model to characterize its essential electronic structure. The associated topological surface states on different surfaces of AsRhTi crystal were also obtained. Even more complicated and exotic multi-loop structure of NLs were predicted in AsFeNb and PNiNb. Our work may shed light on search for exotic multi-loop node-line semimetals in real materials.