Chirality Josephson Current Due to a Novel Quantum Anomaly in Inversion-Asymmetric Weyl Semimetals

We study Josephson junctions based on inversion-asymmetric but time-reversal symmetric Weyl semimetals under the influence of Zeeman fields. We find that, due to distinct spin textures, the Weyl nodes of opposite chirality respond differently to an external magnetic field. Remarkably, a Zeeman field perpendicular to the junction direction results in a phase shift of opposite sign in the current-phase relations of opposite chirality. This leads to a finite chirality Josephson current (CJC) even in the absence of a phase difference across the junction. This feature could allow for applications in chiralitytronics. In the long junction and zero temperature limit, the CJC embodies a novel quantum anomaly of Goldstone bosons at $\pi$ phase difference which is associated with a ${\mathbb{Z}}_2$ symmetry at low energies. It can be detected experimentally via an anomalous Fraunhofer pattern.

Figure 1

(a) Spin texture of the lower conduction band of the model Eq. (1) at $k_y=0$ with ${\kappa }_0=\sqrt{2}$ and $\beta =1$. (b) Position shifts of Weyl nodes in momentum space due to the Zeeman field $h_x$ in $x$ direction. The red (blue) dots denote the Weyl nodes of positive (negative) chirality. (c) Sketch of the Weyl Josephson junction.

Figure 2

Current-phase relations for positive chirality in the absence of Zeeman fields for $L=1/(100\mathrm{\Delta })$ (a) and $1/\mathrm{\Delta }$ (b) with various choices of ${\mu }_N$. (c) Critical current density $J_c^{+}$ as a function of ${\mu }_N$ for $L=1/(100\mathrm{\Delta })$ (i, red line), $1/(10\mathrm{\Delta })$ (ii, blue line), and $1/\mathrm{\Delta }$ (iii, green line), respectively. (d) Location ${\varphi }_{\max }$ of $J_c^{+}$, a measurement of skewness, as a function of ${\mu }_N$ for (i), (ii), and (iii), respectively. ${\mu }_S=100\mathrm{\Delta }$ and $k_{B}T=0.01\mathrm{\Delta }$ in all plots.

Figure 3

Current-phase relations for $h_{x}L=0$ (a), 0.1 (b), $\pi /4$ (c), and $\pi /2$ (d), respectively. The green, yellow, blue, and red curves represent $J_s^{+}$, $J_s^{-}$, $J_s^{\mathrm{tot}}$, and $J_s^{\mathrm{chi}}$, respectively. The inset in panel (d) illustrates the maximum of $J_s^{\mathrm{tot}}$ as a function of $h_{x}L$. $J_0=16\pi \mathrm{\Delta }/(e{R}_n)$, $L=1/(100\mathrm{\Delta })$, ${\mu }_N={\mu }_S=100\mathrm{\Delta }$ and $k_{B}T=0.01\mathrm{\Delta }$ in all plots.

Figure 4

Fraunhofer patterns for $\tilde{g}=0$ (a), 1 (b), 5 (c), and 20 (d), respectively. The blue circled curves are calculated from Eq. (7), while the red curves are plots of Eq. (S5.1) in Ref. [54]. $L=1/(100\mathrm{\Delta })$, ${\mu }_S={\mu }_N=100\mathrm{\Delta }$, and $k_{B}T=0.01\mathrm{\Delta }$ in all plots.