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Topological insulators constitute one of the most intriguing phenomena inmodern condensed matter theory. The unique and exotic properties of topologicalstates of matter allow for unidirectional gapless electron transport andextremely accurate measurements of the Hall conductivity. Recently, newtopological effects occurring at Dirac/Weyl points have been better understoodand demonstrated using artificial materials such as photonic and phononiccrystals, metamaterials and electrical circuits. In comparison, the topologicalproperties of nodal lines, which are one-dimensional degeneracies in momentumspace, remain less explored. Here, we explain the theoretical concept oftopological nodal lines and review recent and ongoing progress using artificialmaterials. The review includes recent demonstrations of non-Abelian topologicalcharges of nodal lines in momentum space and examples of nodal lines realizedin photonic and other systems. Finally, we will address the challenges involvedin both experimental demonstration and theoretical understanding of topologicalnodal lines.

Nodal lines in momentum space: topological invariants and recent realizations in photonic and other systems