Hydrodynamic electron flow in a Weyl semimetal slab: Role of Chern-Simons terms

The hydrodynamic flow of the chiral electron fluid in a Weyl semimetal slab of finite thickness is studied by using the consistent hydrodynamic theory. The latter includes viscous, anomalous, and vortical effects, as well as accounts for dynamical electromagnetism. The energy and momentum separations between the Weyl nodes are taken into account via the topological Chern-Simons contributions in the electric current and charge densities in Maxwell's equations. When an external electric field is applied parallel to the slab, it is found that the electron fluid velocity has a nonuniform profile determined by the viscosity and the no-slip boundary conditions. Most remarkably, the fluid velocity field develops a nonzero component across the slab that gradually dissipates when approaching the surfaces. This abnormal component of the flow arises due to the anomalous Hall voltage induced by the topological Chern-Simons current. Another signature feature of the hydrodynamics in Weyl semimetals is a strong modification of the anomalous Hall current along the slab in the direction perpendicular to the applied electric field. Additionally, it is found that the topological current induces an electric potential difference between the surfaces of the slab that is strongly affected by the hydrodynamic flow.

Figure 1

The schematic band structure of a Weyl semimetal with broken TR and PI symmetries. In equilibrium, the chiral chemical potential ${\mu }_5$ is determined by the separation between the Weyl nodes in energy, i.e., ${\mu }_5=e{b}_0$.

Figure 2

Left panel: The dependence of the fluid velocity components $u_x\left(y\right)$ and $u_y\left(y\right)$ on the spatial coordinate $y$, assuming either the no-slip BCs ($\gamma =1$) or the free-surface ones ($\gamma =0$). Right panel: The electric field $E_y\left(y\right)$ as a function of $y$. We also assumed that $\mathbf{b}\parallel \widehat{\mathbf{z}}$ and used the characteristic values of the parameters in Eq. (59) together with $\mu =10\text{meV}$, ${\mu }_5=0\text{meV}$, $T=10\text{K}$, $E_x=10\mathrm{V}/\mathrm{m}$, $L_y=10\mu \text{m}$, and $C_T={\partial }_{y}T\left(L_y\right)=0$.

Figure 3

The dependence of the electric current density components $J_x\left(y\right)$ (left panel) and $J_y\left(y\right)$ (right panel) on the spatial coordinate $y$, assuming the no-slip BCs ($\gamma =1$) and $\mathbf{b}\parallel \widehat{\mathbf{z}}$. We used the characteristic values of the parameters in Eq. (59) together with $\mu =10\text{meV}$, ${\mu }_5=0\text{meV}$, $T=10\text{K}$, $E_x=10\mathrm{V}/\mathrm{m}$, $L_y=10\mu \text{m}$, and $C_T={\partial }_{y}T\left(L_y\right)=0$.

Figure 4

The dependence of the hydrodynamic AHE current density $J_z(L_y/2)$ (left panel) and the total current per unit length in the $x$ direction $I_z$ (right panel) on $L_y$. The red solid lines correspond to $\mu =10\text{meV}$, the blue dashed lines represent the case $\mu =20\text{meV}$, and the green dotted lines show the results for $\mu =30\text{meV}$. The thin lines correspond to the nonhydrodynamic contributions in $I_z$. We used the parameters in Eq. (59) and set ${\mu }_5=0\text{meV}$, $T=10\text{K}$, $E_x=10\mathrm{V}/\mathrm{m}$, $C_T={\partial }_{y}T\left(L_y\right)=0$, and $\mathbf{b}=\left\{1,0,1\right\}\left|\mathbf{b}\right|/\sqrt{2}$.

Figure 5

The dependence of the hAHE voltage $U$ (left panel) and the hydrodynamic contribution $U_{\mathrm{hydro}}$ (right panel) on $L_y$ at various $T$. The red solid lines correspond to $T=5\text{K}$, the blue dashed lines represent the case $T=10\text{K}$, and the green dotted lines show the results for $T=15\text{K}$. We used the parameters in Eq. (59) and set $\mu =10\text{meV}$, ${\mu }_5=0$, $E_x=10\mathrm{V}/\mathrm{m}$, $C_T={\partial }_{y}T\left(L_y\right)=0$, and $\mathbf{b}\parallel \widehat{\mathbf{z}}$.

Figure 6

The dependence of the normal component of the fluid velocity $u_y(L_y/2)$ and the total electric current per unit length in the $x$ direction $I_z$ on $\mu$ for several values of $T$. The red solid lines correspond to $T=5\text{K}$, the blue dashed lines represent the case $T=10\text{K}$, and the green dotted lines show the results at $T=15\text{K}$. We used the parameters in Eq. (59) and set $\gamma =1$, ${\mu }_5=10\text{meV}$, $E_x=10\mathrm{V}/\mathrm{m}$, $L_y=10\mu \text{m}$, and $C_T={\partial }_{y}T\left(L_y\right)=0$.