Elastic response of the electron fluid in intrinsic graphene: The collisionless regime

The elastic response of an electron fluid at finite frequencies is defined by the electron viscosity $\eta \left(\omega \right)$. We determine $\eta \left(\omega \right)$ for graphene at the charge neutrality point in the collisionless regime, including the leading corrections due to the electron-electron Coulomb interaction. We find interaction corrections to $\eta \left(\omega \right)$ that are significantly larger if compared to the corresponding corrections to the optical conductivity. In addition, we find comparable contributions to the dynamic momentum flux due to single-particle and many-particle effects. We also demonstrate that $\eta \left(\omega \right)$ is directly related to the nonlocal energy-flow response of graphene at the Dirac point. The viscosity in the collisionless regime is determined with the help of the strain generators in the Kubo formalism. Here, the pseudospin of graphene describing its two sublattices plays an important role in obtaining a viscosity tensor that fulfills the symmetry properties of a rotationally symmetric system.

Figure 1

Main: Frequency-dependent shear viscosity, Eq. (6), in pure graphene at charge neutrality and temperature $T=0$ in the high-frequency collisionless regime. The inset: Frequency-dependent shear viscosity in the low-frequency hydrodynamic regime for $T=100\mathrm{K}$ and an inverse scattering time of $\hslash {\tau }_{ee}^{-1}\simeq 1.307{\alpha }_0^2{k}_{B}T\simeq 0.002\mathrm{eV}$ obtained by generalizing the analysis of Ref. [3] to small but finite frequencies [36].

Figure 2

The effect of electron-electron interaction on the shear viscosity of pure graphene at charge neutrality in the high-frequency collisionless regime. The green, dashed lines shows ${\eta }^{\left(0\right)}\left(\omega \right)/{\left(\hslash \omega \right)}^2$ for free Dirac fermions, see Eq. (61). The red curve shows our result (6) for the dynamical shear viscosity $\eta \left(\omega \right)/{\left(\hslash \omega \right)}^2$, including the interaction effects also shown in Fig. 1.

Figure 3

Diagrammatic representation of the stress tensor vertex. The left vertex describes the noninteracting stress tensor (60). The right vertex describes the interaction part of the stress tensor (66).

Figure 4

Feynman diagrams for the correlation function of the stress tensor. Diagram (a) describes the dynamic viscosity of noninteracting Dirac fermions, whereas the remaining diagrams determine the leading-order correction due to the Coulomb interaction. In the main text diagrams (b) and (c) are referred to as the self-energy diagrams, diagram (d)—the vertex diagram, and diagram (e) as the “honey diagram” that includes the interaction correction to the stress tensor.

Figure 5

The honey diagram.