Shear viscosity at the Ising-nematic quantum critical point in two-dimensional metals

In an isotropic strongly interacting quantum liquid without quasiparticles, general scaling arguments imply that the dimensionless ratio $(k_B/\hslash )\eta /s$, where $\eta$ is the shear viscosity and $s$ is the entropy density, is a universal number. We compute the shear viscosity of the Ising-nematic critical point of metals in spatial dimension $d=2$ by an expansion below $d=5/2$. The anisotropy associated with directions parallel and normal to the Fermi surface leads to a violation of the scaling expectations: $\eta$ scales in the same manner as a chiral conductivity, and the ratio $\eta /s$ diverges at low temperature ($T$) as $T^{-2/z}$, where $z$ is the dynamic critical exponent for fermionic excitations dispersing normal to the Fermi surface.

Figure 1

Fermi surface and definition of the momentum components parallel ($k_{\parallel }$) and perpendicular ($k_{\perp }$) to the Fermi surface at the two Fermi surface patches in which the low-energy field theory is defined.

Figure 2

Feynman diagrams yielding the renormalization of the scaling behavior of the viscosity at lowest order in $\epsilon$: (a) One-loop contribution, (b) self-energy correction, and (c) vertex correction. Lines represent fermionic propagators, wiggly lines bosonic propagators, and curly lines the stress tensor.

Figure 3

Elementary excitations due to the chiral electric field that carry a net chiral current at zero total momentum relative to the filled band in a two-patch system. The chiral current can decay via the emission of bosons of opposite momenta on the two patches. Since the individual bosons carry nonzero momentum, the boson distribution responds to the applied chiral electric field and is no longer in equilibrium unlike in a particle-hole symmetric system like those studied in Refs. [56, 58], where the bosons required to relax the elementary excitations have zero momentum.

Figure 4

Transformation of coordinates used to determine the contribution of different patches to the $T_{xy}-T_{xy}$ correlator.