  The hyperscaling property implies that spatially isotropic critical quantum
states in $d$ spatial dimensions have a specific heat which scales with
temperature as $T^{d/z}$, and an optical conductivity which scales with
frequency as $\omega^{(d-2)/z}$ for $\omega \gg T$, where $z$ is the dynamic
critical exponent. We examine the spin-density-wave critical fixed point of
metals in $d=2$ found by Sur and Lee (Phys. Rev. B 91, 125136 (2015)) in an
expansion in $\epsilon = 3-d$. We find that the contributions of the "hot
spots" on the Fermi surface to the optical conductivity and specific heat obey
hyperscaling (up to logarithms), and agree with the results of the large $N$
analysis of the optical conductivity by Hartnoll et al. (Phys. Rev. B 84,
125115 (2011)). With a small bare velocity of the boson associated with the
spin density wave order, there is an intermediate energy regime where
hyperscaling is violated with $d \rightarrow d_t$, where $d_t = 1$ is the
number of dimensions transverse to the Fermi surface. We also present a
Boltzmann equation analysis which indicates that the hot spot contribution to
the DC conductivity has the same scaling as the optical conductivity, with $T$
replacing $\omega$.
