Topological nodal line semimetals is formed when the symmetry of a semimetal system enforces band touching occurring on 1-dimensional submanifolds of the Brillouin Zone. Following the topological classification scheme, various theoretical studies have illuminated that nodal lines can be classified according to $Z_2$-invariants e.g. $\pi$-Berry phase and $Z_2$-monopole charges, though the details depend on the dimensionality and symmetry group of the given system.

But what is the difference between topologically trivial nodal lines and nontrivial ones, in the physical viewpoint?

In the case of Weyl nodes, for example, since they are sources and sinks of berry flux, a section of the BZ can be regarded as a Chern insulator whose Chern number is determined by the flux. Therefore, the charge carried by a Weyl node has physical significance about the transport behavior of the system.

Do $Z_2$-invariants of nodal lines have connection with physical quantities, just as Chern number does with Hall conductance?


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