Figure 1

$PT$ protected nodal lines in (a) and (b), $k·p$ order enriched nodal structures in (c). In (a), for the $k·p$ model $H_{2026}=[R_4(q_y^2-q_x^2)+R_5{q}_x{q}_y]{\sigma }_x+[R_6{q}_z+R_7{q}_z^2+R_8(q_x^2+q_y^2)]{\sigma }_z$, whose CR-required band splitting pattern can only be L-NP, we choose the parameters $R_4=R_5=0.1R_6=R_7=R_8=0.1$. The nodal lines for this $k·p$ model can be visualized the intersection of two energy surfaces. The orange surface is $0.1(q_y^2-q_x^2)+0.1q_x{q}_y=0$ while the green surface is $(q_z+0.1q_z^2)+0.1(q_x^2+q_y^2)=0$ and their intersection contains two nodal lines (in red) which are pinned at the BN located at the origin. In (b), for the $k·p$ model $H_{2373}=[R_4(q_y^2-q_x^2)+R_5{q}_x{q}_z+R_3{q}_y{q}_z+R_6{q}_x{q}_y]{\sigma }_x+(-R_3{q}_z{q}_x+2R_4{q}_x{q}_y+R_5{q}_y{q}_z+R_6\frac{q_x^2-q_y^2}{2}){\sigma }_y$, whose CR-required band splitting pattern can be P-NL, we choose all the parameters to be 1. The orange surface is $q_y^2-q_x^2+q_x{q}_z+q_y{q}_z+q_x{q}_y=0$ while the green surface is $-q_z{q}_x+2q_x{q}_y+q_y{q}_z+\frac{q_x^2-q_y^2}{2}=0$. Their intersection contains one nodal line (in blue) along the $q_z$ axis which is required by CR-required band splitting pattern and the other three nodal lines (in red) not required by CR-required band splitting pattern. (c) demonstrates the $k·p$ order enriched nodal structures using the $k·p$ model $H_{2469}$: The six nodal lines in red, lying in six high-symmetry planes (in gray) are required by CR-required band splitting pattern (the CR-required band splitting pattern is L-NL) but they only appear when the $k·p$ order is 6 or larger. The cyan plane is along $q_x-q_y$ plane which is the solution for nodes when the $k·p$ order takes 1. The yellow surface is the solution for nodes when the $k·p$ order is 2. When the $k·p$ order is 3, 4, or 5, the yellow surface is almost still the node solution. The parameters we choose for the demonstration are shown in Sec. S8 of Supplemental Material [51].