Quasitopological electromagnetic response of line-node semimetals

Topological semimetals are gapless states of matter which have robust surface states and characteristic electromagnetic responses. In this paper, we consider the electromagnetic response of gapless phases in $3+1$ dimensions with line nodes. We show through a layering approach that an intrinsic antisymmetric tensor ${\mathcal{B}}_{\mu \nu }$ (2-form), which is determined by the geometry and energy-embedding of the nodal lines, emerges in the effective response field theory. ${\mathcal{B}}_{\mu \nu }$ is shown to be simply related to the charge polarization and orbital magnetization of the sample; hence the geometry of the line nodes determines these electromagnetic observables. We conclude by discussing the relevance for recently proposed materials and heterostructures with line-node fermi-surfaces.

Figure 1

(a) The polarization for the model in Eq. (2) is plotted vs the parameter $m$ in the model with $\beta =\gamma =2$. (b) The magnetization for the model in Eq. (2) with the extra term $t_{pp}sin{k}_y\mathbb{I}$ is plotted vs various values of $m$ for $\beta =\gamma =2$. (c) The location of the line node is plotted in the $E-k_y-k_z$ space with $k_x=0$ for $t_{pp}=0$ and various values of $m$ with $\beta =\gamma =2$. The polarization is proportional to the area enclosed by the FL. (d) The location of the line node is plotted in the $E-k_y-k_z$ space with $k_x=0$ for $m=1$ and various values of $t_{pp}$ with $\beta =\gamma =2$. The magnetization is proportional to the integral of the energy around the FL in momentum space.

Figure 2

Dotted yellow lines represent initial fourfold degenerate Fermi line $\left(S0\right)$. Purple and blue solid lines represent spin-split Fermi surfaces $(S+,S-)$ with (a) a majority and minority spin Fermi line induced by certain $\mathcal{T}$-breaking terms (b) spin-split Fermi lines with equal sizes for each spin reminiscent of a Rashba-type splitting from spin-orbit terms induced by strain/inversion breaking. For both panels the gray shaded region represents the magnitude of the polarization in the $x$ direction from the projected areas of the Fermi lines after $Z_2$ overlap cancellation.

Figure 3

Rules for the modification of $\chi \text{sgn}{m}_{\mathcal{I}}$ for the determination of the boundary charge for the case of two FLs are illustrated. The green shaded areas represent regions where edge states exist, and the dark green area represents areas where there are overlapping edge states. Case (a) needs a reassignment of arrows while case (b) does not.