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Understanding optical conductivity data in the optimally doped cuprates inthe framework of quantum criticality requires a strongly-coupled quantumcritical 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, thespecific heat scales with temperature as $T^{(d-\theta)/z}$, and the opticalconductivity scales with frequency as $\omega^{(d-\theta-2)/z}$ for $\omega \ggT$, where $z$ is the dynamic critical exponent. We study the Ising-nematiccritical point, using the controlled dimensional regularization method proposedby Dalidovich and Lee (Phys. Rev. B {\bf 88}, 245106 (2013)). We find thathyperscaling is violated, with $\theta =1$ in $d=2$. We expect that similarresults apply to Fermi surfaces coupled to gauge fields in $d=2$.

Hyperscaling violation at the Ising-nematic quantum critical point in two dimensional metals