Anomalous Hall optical conductivity in tilted topological nodal-line semimetals

Nodal-line semimetals are typically characterized by a nodal ring, a closed line of Dirac points in the Brillouin zone, whose topology is characterized by a quantized Berry phase. The nodal loop is protected by the combined inversion and time-reversal ($\mathcal{PT}$) symmetry in the absence of spin-orbit coupling, exhibiting the parity anomaly. Introduction of the $\mathcal{PT}$-break mass term can lead to the transverse Hall response at each point in the nodal ring, but the net Hall current vanishes due to the existence of inversion-symmetry points that contribute to the transverse current with opposite signs. We show that the combination of an inversion-broken tilt term and finite Fermi energy can cause the nonzero anomalous Hall effect. We explore the DC and AC anomalous Hall conductivity in tilted nodal-line semimetals using Kubo formula and focus on the contribution of the free carriers to the Hall conductivity. We find that the interband transition of free carriers dominates the nonvanishing dynamic Hall conductivity over the filled band contribution. The resulting anomalous Hall conductivity exhibits the parity anomaly for the DC case, and rich characteristic frequencies for the AC case which is closely related to the change of the geometry of the Fermi surface. These features may serve as a diagnostic tool to characterize the topological property, tilting parameter, or the radius of the nodal line.

Figure 1

Geometries of Fermi surfaces with blue (red) indicating the electron (hole) pocket in three cases: (a) Dupin cyclide geometry with $u_F>0$ and $v_t=0$; (b) symmetric horn cyclide with $u_F=0$ and $v_t\ne 0$; and (c) asymmetric horn cyclide with $v_t>u_F>0$. The azimuthal angles ${\theta }_{+}$ (${\theta }_{-}$) shows the boundary of the hole (electron) pocket in $k_x-k_y$ plane. (d)–(f) Cross-sectional views of (a)–(c) along the tilt direction with opposite momentum $-\mathbf{k}$ ($\theta =\pi$) and $\mathbf{k}$ ($\theta =0$), respectively. In (e), the tilt effectively modifies the chemical potential $u_F$ to be $u_F-v_t{k}_{\rho }cos\left(\theta \right)$, dependent on the azimuthal angle $\theta$, and so in (f) the hole pocket is suppressed while the electron pocket is enhanced.

Figure 2

The occupation factor $F(k_{\rho }=0.02,k_z=0)$ and the corresponding DC transversal conductivity $\mathrm{Re}\left({\sigma }_{zx}^{\text{free}}\right)$ as a function of (a) and (d) tilt parameter $v_t$, (b) and (e) Fermi energy $u_F$, and (c) and (f) mass term $m$. Finite mass $m=0.05$ is considered in (a), (b), (d), and (e), and $\hslash \omega =0, v{k}_0=1, {\sigma }_0=e^2{v}^2/\left(8\hslash \right)$.

Figure 3

The (a) real and (b) imaginary parts of dynamical Hall conductivity with parameters $v_t=0.3, u_F=0.1$, and $m=0.05$. The solid (dashed) line shows the case of $k_0=1$ ($k_0=0$).

Figure 4

(a) The dispersion in the $k_x$ axis ($k_y=k_z=0$) with finite tilt. Four characteristic frequencies ${\omega }_{1–4}$ characterizing interband transitions are highlighted by red double arrows. (b) The transversal conductivity with a fixed azimuthal angle $\theta =0$ for different $m$ with $v_t=0.4$ and $u_F=0.1$.

Figure 5

The (a) real and (b) imaginary part of ${\sigma }_{zx}^{\text{free}}$ as a function of $\hslash \omega$ for different tilt strength $v_t$ and the Fermi energy $u_F$, respectively, with constant $m=0.05$.

Figure 6

Kerr spectra in the tilted nodal-line semimetal. The relative permittivity is chosen as ${\epsilon }_b=1.5$ and other parameters are set as $v_t=0.3, u_F=0.1, m=0.05$, and $k_0=1$.