Dirac points of electron energy spectrum, band-contact lines, and electron topological transitions of $3\frac{1}{2}$ kind in three-dimensional metals

In many three-dimensional metals with inversion symmetry and a weak spin-orbit interaction, Dirac points of the electron energy spectrum form band-contact lines in the Brillouin zones of these crystals, and electron topological transitions of $3\frac{1}{2}$ kind are due to certain points on these lines. We theoretically study these transitions in detail and point out that they can be detected with magnetic susceptibility which exhibits a giant diamagnetic anomaly at the $3\frac{1}{2}$-order transitions.

Figure 1

The Fermi surfaces near ${\varepsilon }_c={\varepsilon }_{\min }$ at ${\tilde{a}}^2<1$ (here $\tilde{a}=0$): (a) ${\varepsilon }_F<{\varepsilon }_{\min }$ and (b) ${\varepsilon }_F>{\varepsilon }_{\min }$. The band-contact line is shown by the dashed line. If ${\varepsilon }_c-{\varepsilon }_{\min }$ increases, the two conical points in panel (b) become widely separated. The inset shows the FS near one of these points when ${\varepsilon }_{\min }<{\varepsilon }_F<{\varepsilon }_{\max }$, and ${\varepsilon }_F$ is far away from these critical energies.

Figure 2

The Fermi surfaces near ${\varepsilon }_c={\varepsilon }_{\max }$ at $0<{\tilde{a}}^2<1$: (a) ${\varepsilon }_F<{\varepsilon }_{\max }$ and (b) ${\varepsilon }_F>{\varepsilon }_{\max }$. The band-contact line is shown by the dashed line.

Figure 3

The Fermi surfaces near ${\varepsilon }_c={\varepsilon }_{\min }$ at ${\tilde{a}}^2>1$: (a) ${\varepsilon }_F<{\varepsilon }_{\min }$ and (b) ${\varepsilon }_F>{\varepsilon }_{\min }$. The band-contact line is shown by the dashed line.