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 theta, so that in a spatially isotropic state in d spatial dimensions, the specific heat scales with temperature as T(d-theta)/z, and the optical conductivity scales with frequency as omega((d-theta-2)/z) for omega >> 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)]. We find that hyperscaling is violated, with theta = 1 in d = 2. We expect that similar results apply to Fermi surfaces coupled to gauge fields in d = 2.