Graphene: A Nearly Perfect Fluid

Hydrodynamics and collision-dominated transport are crucial to understand the slow dynamics of many correlated quantum liquids. The ratio $\eta /s$ of the shear viscosity $\eta$ to the entropy density $s$ is uniquely suited to determine how strongly the excitations in a quantum fluid interact. We determine $\eta /s$ in clean undoped graphene using a quantum kinetic theory. As a result of the quantum criticality of this system the ratio is smaller than in many other correlated quantum liquids and, interestingly, comes close to a lower bound conjectured in the context of the quark gluon plasma. We discuss possible consequences of the low viscosity, including preturbulent current flow.

Figure 1

(a) Velocity profile $\mathbf{u}$ and associated Stokes force density ${\mathbf{f}}^{\mathbf{s}}=\eta {\nabla }^2\mathbf{u}$ counteracting the current flow. (b) Inhomogeneous current flow expected in a four-contact geometry with split source and drain contacts held at voltage $\pm V/2$. In the absence of viscous and other nonlocal effects, the current would be proportional to the applied voltage $V$, independent of the distance $L$ between the contacts. Viscous effects diminish the current as $L$ decreases.

Figure 2

Ratio $\eta /s$ in graphene as a function of $T$. The UV cut-off was taken to be $T_{\Lambda }=8.34\times {10}^4\mathrm{K}$ following [4]. In the undoped, quantum critical system $\eta /s$ is very small in a large temperature window where the coupling $\alpha (T)$ remains of order $O(1)$. The value $1/4\pi$ obtained for some strongly coupled critical theories is shown for comparison. As shown in the inset, away from zero doping and quantum criticality ($T<|\mu |$), the viscosity assumes the behavior of a degenerate Fermi liquid, $\eta /s\sim (|\mu |/T{)}^3$.