Topological Hopf and Chain Link Semimetal States and Their Application to ${\mathrm{Co}}_2\mathrm{Mn}\text{G}\text{a}$

Topological semimetals can be classified by the connectivity and dimensionality of the band crossings in momentum space. The band crossings of a Dirac, Weyl, or an unconventional fermion semimetal are zero-dimensional (0D) points, whereas the band crossings of a nodal-line semimetal are one-dimensional (1D) closed loops. Here we propose that the presence of perpendicular crystalline mirror planes can protect three-dimensional (3D) band crossings characterized by nontrivial links such as a Hopf link or a coupled chain, giving rise to a variety of new types of topological semimetals. We show that the nontrivial winding number protects topological surface states distinct from those in previously known topological semimetals with a vanishing spin-orbit interaction. We also show that these nontrivial links can be engineered by tuning the mirror eigenvalues associated with the perpendicular mirror planes. Using first-principles band structure calculations, we predict the ferromagnetic full Heusler compound ${\mathrm{Co}}_2\mathrm{MnGa}$ as a candidate. Both Hopf link and chainlike bulk band crossings and unconventional topological surface states are identified.

Figure 1

A variety of new topological semimetals characterized by 3D nontrivial links. (a) Two independent nodal lines protected by two perpendicular mirror planes. (b) When each nodal line passes through the center of the other, the two nodal lines form a nontrivial structure, a Hopf link. A Hopf link is a 3D structure which cannot be mapped to any 2D surface. (c),(d) When the two nodal lines touch at a point, they form a chain. When the two nodal lines are on opposite (same) sides of the touching point, they form an outer (inner) nodal chain.

Figure 2

Mirror eigenvalues and topological surface states. (a) Two independent nodal lines (red and blue) on two perpendicular mirror planes ($M_z$ and $M_x$). The panel below shows the mirror eigenvalues ([$M_x$,$M_z$]) of the VBs and CBs along the $k_y$ axis. We also define a set of $\mathcal{Z}$ numbers $[{\mathcal{Z}}_x,{\mathcal{Z}}_z]$ for the two VBs. ${\mathcal{Z}}_x$ (${\mathcal{Z}}_z$) describes the number of VBs with a positive mirror eigenvalue ($+$) on the ${\mathcal{M}}_x$ (${\mathcal{M}}_z$) plane. (b),(c) The same as (a) but for a Hopf link (b) and an outer or inner nodal chain (c). (d),(e) Surface projections of two independent nodal lines (d) and a Hopf link (e). (f) Independent topological drum-head topological surface states of the two independent nodal lines. (g) Coupled topological drum-head surface states of a Hopf link.

Figure 3

Hopf links and nodal chains in ${\mathrm{Co}}_2\mathrm{MnGa}$. (a) Crystal structure of ${\mathrm{Co}}_2\mathrm{MnGa}$. (b) Spin-down band structure. (c) Spin-up band structure. A band crossing is observed near the $K$ point. (d) Band crossings between bands 1 and 2 all over the bulk BZ. Three types of nodal lines are observed, which are colored by blue, red, and yellow, respectively. The yellow and red nodal lines lie on the $M_x$, $M_y$, and $M_z$ mirror planes. The blue nodal lines lie on the $M_{xy}$, $M_{x\overline{y}}$, $M_{yz}$, $M_{y\overline{z}}$, $M_{xz}$, and $M_{x\overline{z}}$ mirror planes. (e) The red and blue nodal lines at the corner of the Brillouin zone. Y1 and R1 (Y2 and R2) are located on the $M_z$ ($M_x$) mirror plane. (f) 3D band structure ($E-k_x-k_z$) of a blue nodal line. (g)–(i) Decomposing the complicated nodal features into the basis units in Fig. 1. (g) Two yellow nodal lines form a Hopf link. (h) A blue nodal line and a red nodal line (or a yellow and a red nodal line) form an outer nodal chain. (i) A blue nodal line and a yellow nodal line form an inner nodal chain.

Figure 4

Drum-head topological surface states in ${\mathrm{Co}}_2\mathrm{MnGa}$. (a) The bulk BZ and the projection of all nodal features onto the (001) surface. (b) Surface states with Co termination at 100 meV below the Fermi level. The sharp features are surface states, whereas the shaded region are projections of bulk bands. (c) Schematic illustration of the surface state connectivity observed in (b). The green curves represent the surface states, whereas other lines are the projections of the bulk nodal lines as in (a). (d),(e) Energy dispersions along the white and red cuts indicated in (c). Surface states connecting adjacent band crossings are observed.