Simple model for second-order topological insulators and loop-nodal semimetals in Transition Metal Dichalcogenides XTe$_2$ (X=Mo,W)
Transition metal dichalcogenides XTe$_{2}$ (X=Mo,W) have been shown to be second-order topological insulators based on first-principles calculations, while topological hinge states have been shown to emerge based on the associated tight-binding model. The model is equivalent to the one constructed from a loop-nodal semimetal by adding mass terms and spin-orbit interactions. We propose to study a chiral-symmetric model obtained from the original Hamiltonian by simplifying it but keeping almost identical band structures and topological hinge states. A merit is that we are able to derive various analytic formulas because of chiral symmetry, which enables us to reveal basic topological properties of transition metal dichalcogenides. We find a linked loop structure where a higher linking number (even 8) is realized. We construct second-order topological semimetals and two-dimensional second-order topological insulators based on this model. It is interesting that topological phase transitions occur without gap closing between a topological insulator, a topological crystalline insulator and a second-order topological insulator. We propose to characterize them by symmetry detectors discriminating whether the symmetry is preserved or not. They differentiate topological phases although the symmetry indicators yield identical values to them. We also show that topological hinge states are controllable by the direction of magnetization. When the magnetization points the $z$ direction, the hinges states shift, while they are gapped when it points the in-plane direction. Accordingly, the quantized conductance is switched by controlling the magnetization direction. Our results will be a basis of future topological devices based on transition metal dichalcogenides.