Viscometry of Electron Fluids from Symmetry

When electrons flow as a viscous fluid in anisotropic metals, the reduced symmetry can lead to exotic viscosity tensors with many additional, nonstandard components. We present a viscometry technique that can, in principle, measure the multiple dissipative viscosities allowed in isotropic and anisotropic fluids alike. By applying representation theory to exploit the intrinsic symmetry of the fluid, our viscometry is also exceptionally robust to both boundary complications and ballistic effects. We present the technique via the illustrative example of dihedral symmetry, relevant in this context as the point symmetry of 2D crystals. Finally, we propose a present-day realizable experiment for detecting, in a metal, a novel hydrodynamic phenomenon: the presence of rotational dissipation in an otherwise isotropic fluid.

Figure 1

Illustration of the origin of rotational viscosity in electron fluids. When an anisotropic Fermi surface (black) is rotated (dark purple), quasiparticle excitations (red, blue) are generated. In the hydrodynamic limit, such rigid rotations are opposed by a dissipative rotational viscosity ${\eta }_{\circ }$ [21]. Note that this Fermi surface has ${\mathsf{D}}_8$ symmetry.

Figure 2

First row: the five irreducible representations of ${\mathsf{D}}_8$. Second row: current boundary conditions (blue and red arrows) of matching ${\mathsf{D}}_8$ symmetry, indicated by colored wedges. Symmetry restricts heat (5) at the square center to only a single dissipative coefficient (yellow disk). Note that the representation $U_0^{+}$ requires more than eight contacts in order to satisfy charge conservation.

Figure 3

Flows numerically solving Eq. (3) in our viscometer with $w=1\mu \mathrm{m}$, $I_0=100\mu \mathrm{A}$, $d/w=0.41$, $a/w=0.05$, $\delta =0$, and $\lambda /w=\infty$. Rows specify ${\mathsf{D}}_8$-irreducible boundary conditions, and columns give the temperature variation $-({\nabla }^{2}T{)}_{\alpha }$ sourced solely by ${\eta }_{\alpha }$ dissipation. Symmetry restricts center heating to only the diagonal plots. In giving an order-of-magnitude estimate for the scale of heating, we have taken relevant physical parameters from hydrodynamic electrons in monolayer graphene [6, 7]; see the Supplemental Material [34]. Temperature variations of this magnitude are detectable with existing local thermometers [47, 48].

Figure 4

Viscometer center heat $W_{\mathbf{0}}$, numerically determined from Eq. (3), as a function of boundary condition irrep, contact spacing $d$, and anisotropy $\delta$, for $a/w=0.01$ and $\lambda /w=\infty$. Each curve is normalized by its maximum value. The uniqueness of these curves should allow for experimental determination of $\delta$. Although momentum relaxation is neglected in these $\lambda /w=\infty$ plots, we find that the shape of these curves, and hence their utility in determining $\delta$, is extremely insensitive to decreasing $\lambda$ (increasing $\mathrm{\Gamma }$); see the Supplemental Material [34].