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 $\omega \gg T$, 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 $d=2$.

Figure 1

Feynman diagrams for the contributions to the current-current correlation function in (a) $\mathcal{O}\left(N\right)$ and (b),(c) $\mathcal{O}\left(1\right)$. The wiggly line represents the bosonic propagator and the curly line the vector potential.

Figure 2

Feynman diagram for the two-loop interaction correction to the free energy.

Figure 3

Diagrammatic representation of the one-loop contributions to (a) the fermionic self-energy and (b) the current vertex.