Violation of hyperscaling at the Ising-nematic quantum critical point in a two-dimensional metal

Spatially isotropic critical quantum states in $d$ spatial dimensions which have the hyperscaling property have an optical conductivity that scales as $\omega^{(d-2)/z}$ for high frequencies $\omega >> T$, where $T$ is the temperature and $z$ the dynamic critical exponent. We examine the Ising-nematic quantum critical point in $d = 2$ using the fixed point theory by Dalidovich and Lee (Phys. Rev. B 88, 245106 (2013)) and compute the optical conductivity in an expansion in $\epsilon = 5/2 - d$. We show that hyperscaling is violated at this quantum critical point and discuss the scaling behaviour of the optical conductivity at $T = 0$.