Dissipative and Hall Viscosity of a Disordered 2D Electron Gas

Hydrodynamic charge transport is at the center of recent research efforts. Of particular interest is the nondissipative Hall viscosity, which conveys topological information in clean gapped systems. The prevalence of disorder in the real world calls for a study of its effect on viscosity. Here we address this question, both analytically and numerically, in the context of disordered noninteracting 2D electrons. Analytically, we employ the self-consistent Born approximation, explicitly taking into account the modification of the single-particle density of states and the elastic transport time due to the Landau quantization. The reported results interpolate smoothly between the limiting cases of a weak (strong) magnetic field and strong (weak) disorder. In the regime of a weak magnetic field our results describe the quantum (Shubnikov–de Haas type) oscillations of the dissipative and Hall viscosity. For strong magnetic fields we characterize the effects of the disorder-induced broadening of the Landau levels on the viscosity coefficients. This is supplemented by numerical calculations for a few filled Landau levels. Our results show that the Hall viscosity is surprisingly robust to disorder.

Figure 1

(a) The self-energy diagram. (b) The diagram corresponding to the Kubo formula (2). (c) The equation for the vertex $V_m$ in the ladder approximation. The solid line denotes the SCBA Green’s function $\mathcal{G}(ϵ)$. The dashed line denotes the disorder correlation function $W(\mathit{r})$.

Figure 2

The density of states (blue dotted and blue dash-dotted curves) and shear viscosity (red solid and red dashed curves) as functions of $\mu /{\omega }_c$ for smooth disorder, ${\tau }_{\mathrm{tr},2}/{\tau }_0=40$. Blue dotted and red solid curves correspond to well-separated LLs with ${\omega }_c{\tau }_0=15$. Blue dash-dotted and red dashed curves correspond to overlapping LLs with ${\omega }_c{\tau }_0=0.8$.

Figure 3

The normalized Hall viscosity $8\pi l_B^2{\eta }_H/N^2$ as the function of $\mathrm{\Gamma }/{\omega }_c$ for different ranges of disorder in the case of well-separated LLs. The parameter ${\tau }_{\mathrm{tr},2}/{\tau }_0$ is equal to 1, 2, 3, and 10 from the bottom to the top.

Figure 4

The Hall conductivity (circles) and viscosity (squares) as a function of ${\omega }_c\tau$ for $N=0$, 1, and 2 filled Landau levels (blue, yellow, and green, respectively). The clean viscosity values are also indicated by dashed lines. ${\eta }_H$ is seen to be robust to disorder to the same extent the ${\sigma }_H$ is, within the numerical errors (error bars are smaller than the symbol sizes).