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 nonzero 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 ≫T , where z is the dynamic critical exponent defined by the scaling of the fermion response function transverse to the Fermi surface. We study the Ising-nematic critical point, using the controlled dimensional regularization method proposed by Dalidovich and Lee [Phys. Rev. B 88, 245106 (2013), 10.1103/PhysRevB.88.245106]. 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 .