Charge and Valley Hydrodynamics in the Quantum Hall Regime of Gapped Graphene

Abstract

We develop a unified viscous hydrodynamics for charge and valley transport in gapped graphene in the quantum Hall regime. We redefine Hall viscosity as a response to static electric-field gradients instead of strain, establishing a derivative hierarchy that fundamentally links it to nonlocal Hall conductivity. The theory predicts quantized Hall viscosity for charge and valley, including a ground-state contribution. Crucially, the valley current is unaffected by the Lorentz force and is directly accessible via the local pressure, namely the electrostatic potential that tracks fluid vorticity.

Figures (9)

• Figure 1: Schematic of the von Neumann lattice and coarse graining. The von Neumann lattice consists of the phase-space unit cells localized at Landau-level guiding centers, occupying an area of $2\pi l_B^2$. The fluid element $\Omega_R$ may be chosen as $\sim l_B W$, which is macroscopically small relative to the device area yet microscopically large compared with the von Neumann lattice unit cell. Throughout the Letter, we work within the scale separation $l_B, l_{\mathrm{ee}} \ll \sqrt{\Omega_R} \ll W,L\ll l_{\mathrm{mfp}}$. Here $l_B=\sqrt{\hbar/(eB)}$ is the magnetic length; $W,L$ is the device width and length; $l_{\mathrm{ee}}$ is the electron–electron scattering length; and $l_{\mathrm{mfp}}$ is the elastic mean free path.
• Figure 2: Magnetic field $B$ dependence of charge $\eta_H^{c}$ and valley $\eta_H^{v}$ Hall viscosity in units of $1/4\pi l_B^2$, with a typical concentration $n$. The distinct zeroth energy level in two valley is represented by $0^\pm$. The inset displays the semi-classical behavior at moderate $B$ following Refs. p.s.alekseevNegativeMagnetoresistanceViscous2016thomasscaffidiHydrodynamicElectronFlow2017, which depends on system details: $l_{\mathrm{ee}},\epsilon_F$, and the effective mass $m^*\equiv\epsilon_F/v_F^2$.
• Figure 3: Numerical solutions of Eq. \ref{['8']} with $\mathrm{Re}=0.2$; device size: $L=10\mathrm{\mu m}, W=4\mathrm{\mu m}$; boundary condition: $j_x(x=0)= v_\text{in}, j_x(y=\pm W/2)=\pm v_\text{in}$. (a) Stream flow for VQHE with $k=0.1$ and $\omega_B=0$. (b) Flow difference between VHE and VQHE (Hall-viscosity effect). (c) Flow difference between VQHE and CQHE (Lorentz-force effect).
• Figure 4: (a) Pressure variation $\delta p$ between VHE and VQHE (Hall-viscosity contribution). (b) Pressure variation $\delta p$ between VQHE and CQHE (Lorentz-force contribution). (c) Random sampling in the bulk used to extract $\widetilde{\eta}_H=1/\mathrm{Re}=5$ (blue points) and $\widetilde{\omega}_B\equiv\omega_B L/v_{\text{in}}=10$ (yellow points).
• Figure S1: Hall conductivity in strong magnetic field $B=14\mathrm{T}$ for gapless and gapped graphene versus Concentration $n$. Compared with gapless graphene, gapped graphene exhibits an additional zero-conductance plateau at filling factor $\nu=0$. Insets show the phase diagram for gapped graphene in both valley, where the conductivity is displayed in unit of $g_se^2/h$. $\gamma\equiv\Delta/\hbar\omega_B$