Irreducible momentum-space spin structure of Weyl semimetals and its signatures in Friedel oscillations

Materials that break time-reversal or inversion symmetry possess nondegenerate electronic bands, which can touch at so-called Weyl points. The spinor eigenstates in the vicinity of a Weyl point exhibit a well-defined chirality $\pm 1$. Numerous works have studied the consequences of this chirality, for example, in unconventional magnetoelectric transport. However, even a Weyl point with isotropic dispersion is not only characterized by its chirality but also by the momentum dependence of the spinor eigenstates. For a single Weyl point, this momentum-space spin structure can be brought into standard “hedgehog” form by a unitary transformation, but for two or more Weyl points, this is not possible. In this work, we show that the relative spin structure of a pair of Weyl points has strong qualitative signatures in the electromagnetic response. Specifically, we investigate the Friedel oscillations in the induced charge density due to a test charge for a centrosymmetric system consisting of two Weyl points with isotropic dispersion. The most pronounced signature is that the amplitude of the Friedel oscillations falls off as $1/r^4$ in directions in which both Weyl points exhibit the same spin structure, while for directions with inverted spin structures, the amplitude of the Friedel oscillations decreases as $1/r^3$.

Figure 1

Top left: Spin structure of the Weyl point with positive chirality for all three models. The spin of the Bloch states on a constant-energy surface surrounding the Weyl point are given by $|\to \rangle =\left(\right|\uparrow \rangle +|\downarrow \rangle )/\sqrt{2}, |\leftarrow \rangle =\left(\right|\uparrow \rangle -|\downarrow \rangle )/\sqrt{2}, |\otimes \rangle =\left(\right|\uparrow \rangle +i|\downarrow \rangle )/\sqrt{2}$, and $|\odot \rangle =\left(\right|\uparrow \rangle -i|\downarrow \rangle )/\sqrt{2}$, where $\left\{\right|\uparrow \rangle ,|\downarrow \rangle \}$ is the eigenbasis of the Pauli matrix ${\sigma }_z$. Top right to bottom right: Spin structures of the Weyl point with negative chirality for models $\alpha =a,b,c$.

Figure 2

Functions $f\left(x\right), u(x,5)$, and $g\left(x\right)$ of $x=q/2k_F$, given in Eqs. (25), (30), and (35), respectively.

Figure 3

Band structure and intrinsic polarizability for the tight-binding model given in Eq. (38) with $\gamma =0$ (black), $\gamma =0.2$ (dark brown), and $\gamma =0.5$ (light brown). (a) Band structure along the $k_z$ axis. (b) Intrinsic polarizability as a function of $q_z$ for $q_x=q_y=0$. (c) Comparison of ${\pi }_{\mathrm{tb}}^{\mathrm{int}}\left(\mathit{q}\right)$ (solid lines) and ${\pi }_{\mathrm{intra}}^{\mathrm{int},\alpha }\left(\mathit{q}\right)$ for $\mathrm{\Lambda }=2$ (triangles), $\mathrm{\Lambda }=1.6$ (stars), and $\mathrm{\Lambda }=1.14$ (plus signs). The remaining parameters for the tight-binding model are $\eta =3\times {10}^{-4}, t_x=t_y=1$, and $t_z=1/\sqrt{1-{\gamma }^2}$. For ${\pi }_{\mathrm{intra}}^{\mathrm{int},\alpha }\left(\mathit{q}\right), \hslash v_F=1$ was chosen.

Figure 4

Second derivatives of the intrinsic polarizability of the tight-binding model with respect to different momentum components for various $\gamma$ and broadenings $\eta$. (a) Second derivative in $q_z$ for $q_x=q_y=0$ and various $\gamma$. The separations between the Weyl points are given by $Q_0=\pi , Q_1=2.74$, and $Q_2=2.09$ for the three values of $\gamma$. (b) For $\gamma =0.5$, comparison of the second derivative in $q_z$ in the vicinity of $Q_2$ along the $q_z$ axis for various $\eta$ with ${\alpha }_{z}ln|q_z-Q_2|$, where ${\alpha }_z=0.0078$. (c) For $\gamma =0.5$, comparison of the second derivative in $q_x$ for $q_y=0$ and $q_z=Q_2$ for various $\eta$ with ${\alpha }_{x}ln\left|q_x\right|$, where ${\alpha }_x=-0.024$. Unless stated otherwise, the used parameters are the same as in Fig. 3.

Figure 5

Induced charge density and frequency spectrum of the Friedel oscillations for model system $a$. (a) Induced charge density for various $\tilde{Q}=Q/2k_F$ along the ${\tilde{r}}_z$ axis (${\tilde{r}}_x={\tilde{r}}_y=0$), scaled with ${\tilde{r}}_z^n$, where $n=4$ for the upper two plots and $n=3$ for the lower plot. The black dashed curve shows the induced charge density including only the extrinsic intravalley polarizability. The effective fine-structure constant is taken to be $r_s=1$. (b)–(d) Fourier components of the discrete Fourier transform of the Friedel oscillations from (a).

Figure 6

Induced charge density for various $\tilde{Q}=Q/2k_F$ along the ${\tilde{r}}_x$ axis for model system $a$. The black dashed curve shows the induced charge density including only the extrinsic intravalley polarizability. The effective fine-structure constant is taken to be $r_s=1$.

Figure 7

Induced charge density for different $\tilde{Q}=Q/\left(2k_F\right)$ for model system $b$ along (a) the ${\tilde{r}}_x$ axis, and (b) the ${\tilde{r}}_z$ axis. The black dashed curve shows the induced charge density including only the extrinsic intravalley polarizability. The effective fine-structure constant is taken to be $r_s=1$.

Figure 8

Illustration of transitions between states at the equators of the Fermi surfaces for model $b$. The transition shown in blue is enhanced due to the spin structure, while the one in red is suppressed.

Figure 9

Induced charge density for model system $c$ along (a) the ${\tilde{r}}_x$ axis for $\tilde{Q}=Q/\left(2k_F\right)=3$, and (b) the ${\tilde{r}}_z$ axis for different $\tilde{Q}$. The black dashed curve shows the induced charge density including only the extrinsic intravalley polarizability. The effective fine-structure constant is taken to be $r_s=1$.

Figure 10

Intervals of the Friedel oscillations in Fig. 5 chosen to calculate the Fourier components in Figs. 5–5.