Plasmon modes in quadratic and cubic nodal line semimetals

Abstract

Nodal line semimetals (NLSMs) host distinctive topological properties and unconventional collective excitations. While plasmons in linear NLSMs are well studied, those of higher-order dispersive NLSMs remain poorly understood. We investigate plasmon modes in 3D quadratic and cubic NLSMs within the random phase approximation (RPA), valid for $r_s\ll 1$ but breaking down in the small-doping cubic regime where the density of states diverges. Deriving the one-loop polarizability and evaluating its coefficients numerically from the full 3D Lindhard integral, we find distinct carrier-density scaling laws: quadratic NLSMs exhibit ${\omega }_p\sim n^{1/2}$, while cubic NLSMs show ${\omega }_p\sim n^{2/3}$ at large doping crossing over to ${\omega }_p\sim n^{3/4}$ at small doping. The $n^{3/4}$ law is not quantitatively reliable: the diverging density of states drives $r_s>1$ already at $n\sim 10^{18}$ cm${}^{-3}$, limiting the observable window to at most $10^{18}\lesssim n\lesssim 10^{19}$ cm${}^{-3}$; below this, beyond-RPA correlations are essential. These scalings originate from the power-law density of states and intraband (Drude) response, and are analogous in phase-space power counting to bilayer and trilayer graphene, respectively. The long-wavelength intraband polarization dominates the plasmon frequency, while interband contributions are subleading (two to three orders of magnitude smaller for $\Omega \ll \mu$). Both systems share a plasmon anisotropy ${\Omega }_p^z/{\Omega }_p^{\perp }\to \sqrt{2}$ in the thin-ring limit, a generic consequence of the torus Fermi-surface geometry. This $\sqrt{2}$ doublet, though observable by HREELS, cannot distinguish quadratic from cubic dispersion; the density scaling exponent is the true distinguishing signature. Our results provide a framework for the collective dynamics of higher-order dispersive NLSMs and suggest experimentally testable signatures accessible by HREELS.