Anomalous Gurzhi effect

We investigate the role of the chiral anomaly in hydrodynamic and crossover regimes of transport in a Weyl or Dirac semimetal film. We show that the magnetic-field-dependent part of the electric conductivity in the direction of the magnetic field develops an unusual nonmonotonic dependence on temperature dubbed the anomalous Gurzhi effect. This effect is realized in sufficiently clean semimetals subject to a classically weak magnetic field where the electron-electron scattering dominates and the relaxation of the valley-imbalance charge density happens at the boundaries. Moreover, the conditions for the realization of the conventional Gurzhi effect in hydrodynamic and crossover regimes of transport in three-dimensional Dirac and Weyl semimetals are determined.

Figure 1

A schematic model setup where static electric $\mathbf{E}$ and magnetic $\mathbf{B}$ fields are applied along the film of a Weyl or Dirac semimetal. The thickness of the film along the $y$ direction is $L$ and the film is infinite along the $x$ and $z$ directions. We force the chiral charge density $N_5\left(y\right)$ to vanish at the surfaces of the film by the boundary conditions $N_5(y=0,L)=0$.

Figure 2

Transport phase diagram of the system with three main regimes: (i) Ohmic (${\lambda }_{\mathrm{G}}\ll L$ and ${\lambda }_{\mathrm{G},5}\ll L$), (ii) chiral hydrodynamic (${\lambda }_{\mathrm{G}}\ll L$ and ${\lambda }_{\mathrm{G},5}\gg L$), and (iii) hydrodynamic (${\lambda }_{\mathrm{G}}\gg L$ and ${\lambda }_{\mathrm{G},5}\gg L$). Here, ${\lambda }_{\mathrm{G}}\sim \sqrt{l_{\mathrm{ee}}{l}_{\mathrm{ei}}}$ and ${\lambda }_{\mathrm{G},5}\sim \sqrt{l_{\mathrm{ee}}{l}_{\mathrm{ei},5}}$ are Gurzhi and chiral Gurzhi lengths, respectively, where $l_{\mathrm{ei}}, l_{\mathrm{ei},5}$, and $l_{\mathrm{ee}}$ are the intranode, internode, and electron-electron scattering lengths, respectively. Since the intranode electron-impurity scattering length is usually smaller than the internode one [42, 43], i.e., $l_{\mathrm{ei}}\ll l_{\mathrm{ei},5}$, the regime ${\lambda }_{\mathrm{G}}\gtrsim {\lambda }_{\mathrm{G},5}$ (gray shaded region) cannot be realized.

Figure 3

Dependence of the resistivity ${\rho }_{\parallel }$ on temperature for a few values of the dimensionless impurity scattering length $l_{\mathrm{ei},0}=l_{\mathrm{ei}}\left(p_F\right)/L$ at ${\alpha }_{\mathrm{eff}}=0.2, {\lambda }_F=0.001L$, and $N_W=24$. A rather small Fermi length is used to emphasize the features of the result. The dashed vertical line corresponds to the temperature at which $l_{\mathrm{ee}}=2L$.

Figure 4

Dependence of the anomalous part of the conductivity ${\sigma }_{\mathrm{anom}}$ [panel (a)] and the relative anomalous contribution to the resistivity $1-{\rho }_{\parallel }\left(B\right)/{\rho }_{\parallel }(B=0)$ [panel (b)] on temperature for several values of the magnetic length $l_B=\sqrt{\hslash c/\left(eB\right)}$. In all panels, we set ${\lambda }_F=0.001L, l_{\mathrm{ei},0}=100, l_{\mathrm{ei},5}/l_{\mathrm{ei}}=100, {\alpha }_{\mathrm{eff}}=0.2$, and $N_W=24$. The dashed vertical line corresponds to the temperature at which $l_{\mathrm{ee}}=2L$.