Tilt-controlled Drude anisotropy in nodal-line semimetals: logarithmic enhancement, interband correction, gauge consistency, and the Landau-damping window

Abstract: We study the anisotropic charge response of a three-dimensional tilted nodal-line semimetal in which the tilt enters the spectrum through an identity term and acts as the model control parameter. Using the Kubo formula with line-integral vertices, we show that the tilt-induced deformation of the low-energy phase space produces parametrically different longitudinal and transverse Drude responses: the longitudinal weight grows logarithmically as $D_{zz}=B\ln(1/δ)$ with the analytically determined coefficient $B=g/(2π)<sup>2$---within the window in which the Fermi-surface tail lies inside the ultraviolet cutoff, while at a fixed physical cutoff the weight saturates instead at a cutoff-set plateau---whereas the transverse weight follows a stronger algebraic scaling $D_\perp\simδ<sup>{-1}$; the leading $\sqrtδ$ corrections to both asymptotic forms are obtained analytically. The interband polarization retains the $q<sup>2$ long-wavelength structure, with a coefficient that is infrared-integrable through Pauli blocking at finite doping yet carries a slow ultraviolet logarithm; it renormalizes the collective mode multiplicatively and sets the interband absorption threshold $ω_c=2μ/(1+|η_z|)$. A self-consistent solution of the plasmon dispersion identifies a tilt-dependent damping-free window bounded by the interband continuum. We predict that the type-II extension of this window, within the stated bandwidth (Brillouin-zone-like) cutoff regularization, contains a candidate low-frequency acoustic-like branch outside the particle--hole continua for $η_z\gtrsim1.2$ over a finite window in momentum transfer.