Dynamo Effect and Turbulence in Hydrodynamic Weyl Metals

The dynamo effect is a class of macroscopic phenomena responsible for generating and maintaining magnetic fields in astrophysical bodies. It hinges on the hydrodynamic three-dimensional motion of conducting gases and plasmas that achieve high hydrodynamic and/or magnetic Reynolds numbers due to the large length scales involved. The existing laboratory experiments modeling dynamos are challenging and involve large apparatuses containing conducting fluids subject to fast helical flows. Here we propose that electronic solid-state materials—in particular, hydrodynamic metals—may serve as an alternative platform to observe some aspects of the dynamo effect. Motivated by recent experimental developments, this Letter focuses on hydrodynamic Weyl semimetals, where the dominant scattering mechanism is due to interactions. We derive Navier-Stokes equations along with equations of magnetohydrodynamics that describe the transport of a Weyl electron-hole plasma appropriate in this regime. We estimate the hydrodynamic and magnetic Reynolds numbers for this system. The latter is a key figure of merit of the dynamo mechanism. We show that it can be relatively large to enable observation of the dynamo-induced magnetic field bootstrap in an experiment. Finally, we generalize the simplest dynamo instability model—the Ponomarenko dynamo—to the case of a hydrodynamic Weyl semimetal and show that the chiral anomaly term reduces the threshold magnetic Reynolds number for the dynamo instability.

Figure 1

Flow regimes for the electron liquid in a Weyl semimetal on the diagram “fine-structure constant” $\alpha =(e^2/\varkappa \hslash v_F)$ vs flow velocity $u$ (log-log scale) for the room temperature $T=T_{\mathrm{room}}=300\mathrm{K}$ and the Fermi velocity $v_F={10}^8(\mathrm{cm}/\mathrm{s})$. The maximum value of the fine-structure constant is ${\alpha }_{\max }=(e^2/\hslash v_F)\approx 2.2$.

Figure 2

The critical magnetic Reynolds number $R_m^c$ of the $n=1$ kinematic Ponomarenko dynamo as a function of the helicity parameter $\delta =4\pi \sigma \xi /k{c}^2$, with $k$ being the wave vector of the dynamo instability. $R_m^c\simeq 17.7$ is the critical value for the dynamo in the absence of helicity. We note that a self-exciting dynamo will always correspond to the chirality with a lower critical Reynolds number. The chiral anomaly, thus, always aids the dynamo effect.