Quadratic and cubic nodal lines stabilized by crystalline symmetry

In electronic band structures, nodal lines may arise when two (or more) bands contact and form a one-dimensional manifold of degeneracy in the Brillouin zone. Around a nodal line, the dispersion for the energy difference between the bands is typically linear in any plane transverse to the line. Here, we explore the possibility of higher-order nodal lines, i.e., lines with higher-order dispersions, that can be stabilized in solid-state systems. We reveal the existence of quadratic and cubic nodal lines, and we show that these are the only possibilities (besides the linear nodal line) that can be protected by crystalline symmetry. We derive effective Hamiltonians to characterize the novel low-energy fermionic excitations for the quadratic and cubic nodal lines, and explicitly construct minimal lattice models to further demonstrate their existence. Their signatures can manifest in a variety of physical properties such as the (joint) density of states, magnetoresponse, transport behavior, and topological surface states. Using ab initio calculations, we also identify possible material candidates that realize these exotic nodal lines.