Hyperscaling at the spin density wave quantum critical point in two dimensional metals

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 $\omega$.