Unified Theory of $PT$ and $CP$ Invariant Topological Metals and Nodal Superconductors

As $PT$ and $CP$ symmetries are fundamental in physics, we establish a unified topological theory of $PT$ and $CP$ invariant metals and nodal superconductors, based on the mathematically rigorous $KO$ theory. Representative models are constructed for all nontrivial topological cases in dimensions $d=1$, 2, and 3, with their exotic physical meanings being elucidated in detail. Intriguingly, it is found that the topological charges of Fermi surfaces in the bulk determine an exotic direction-dependent distribution of topological subgap modes on the boundaries. Furthermore, by constructing an exact bulk-boundary correspondence, we show that the topological Fermi points of the $PT$ and $CP$ invariant classes can appear as gapless modes on the boundary of topological insulators with a certain type of anisotropic crystalline symmetry.

Figure 1

(a) A small sphere (circle) (cyan) enclosing the gapless point (ring) (red) can be deformed into two large spheres (circles) (green) in the BZ. Note that the BZ is periodic in $k$. (b) Lattice model with topological Fermi points (red) in its bulk BZ (light green). The boundary BZs for the (11) and (01) edges are indicated by blue lines.