Abstract

Understanding optical conductivity data in the optimally doped cuprates
in the framework of quantum criticality requires a strongly-coupled
quantum critical metal which violates hyperscaling. In the simplest
scaling framework, hyperscaling violation can be characterized by a
single non-zero exponent $\theta$, so that in a spatially isotropic
state in $d$ spatial dimensions, the specific heat scales with
temperature as $T^{(d-\theta)/z}$, and the optical conductivity scales
with frequency as $\omega^{(d-\theta-z)/z}$ for $\omega \gg T$, where
$z$ is the dynamic critical exponent for fermionic excitations
dispersing normal to the Fermi surface. We study the Ising-nematic
quantum critical point using dimensional regularization and find that
hyperscaling is violated with $\theta = 1$ in $d = 2$. This result can
be traced back to the anisotropic scaling of momenta in the directions
parallel and normal to the Fermi surface and an emergent symmetry of the
fixed point theory. Moreover, due to this anisotropy, the ratio $\eta /
s$ between the shear viscosity $\eta$ and the entropy density $s$
diverges at low temperatures as $T^{-2/z}$, instead of approaching a
universal number as expected for a strongly interacting quantum liquid
without quasiparticles.