Logarithmically enhanced intrinsic nonlinear Hall effects in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetric antiferromagnetic nodal-line semimetals

We investigate the intrinsic nonlinear Hall effect (INHE) in $\mathcal{PT}$-symmetric antiferromagnetic nodal-line semimetals, highlighting a significant enhancement characterized by a logarithmic divergence in the INH conductivity as the Fermi energy intersects the nodal line within a finite energy range. Through analytical derivations and numerical calculations, we demonstrate that the INH conductivity logarithmically depends on the gap opening at the nodal line. We further reveal that current-induced spin-orbit torque, which rotates the N\'eel vector and opens a gap at the nodal line, results in a nonanalytical transverse response, where the amplitude of high-harmonic generation decreases as the inverse cube of the harmonic order. Our finding not only serves as a pathway to an enhanced nonlinear Hall response without the necessity of fine tuning the chemical potential but also suggests a different nonanalytical response that may contribute to the experiment identification of topological nodal-line semimetals and potential applications in ultrafast nonlinear devices.