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

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 $θ$, so that in a spatially isotropic state in $d$ spatial dimensions, the specific heat scales with temperature as $T^{(d-θ)/z}$, and the optical conductivity scales with frequency as $ω^{(d-θ-2)/z}$ for $ω\gg T$, where $z$ is the dynamic critical exponent. We study the Ising-nematic critical point, using the controlled dimensional regularization method proposed by Dalidovich and Lee (Phys. Rev. B {\bf 88}, 245106 (2013)). We find that hyperscaling is violated, with $θ=1$ in $d=2$. We expect that similar results apply to Fermi surfaces coupled to gauge fields in $d=2$.

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