Electron self-energy near a nematic quantum critical point

We consider an isotropic Fermi liquid in two dimensions near the n=2 Pomeranchuk instability in the charge channel. The order parameter is a quadrupolar stress tensor with two bosonic shear modes with polarizations longitudinal and transverse to the quadrupolar momentum tensor. Longitudinal and transverse bosonic modes are characterized by dynamical exponents z∥=3 and z⊥ =2, respectively. Previous studies have found that such a system exhibits multiscale quantum criticality with two different energy scales ω∼ ξ^-z∥,⊥, where ξ is the correlation length. We study the impact of the multiple energy scales on the electron Green's function. The interaction with the critical z∥ =3 mode is known to give rise to a local self-energy that develops a non-Fermi-liquid form, Σ (ω) ∼ ω^2/3 for frequencies larger than the energy scale ω∼ ξ^-3. We find that the exchange of transverse z⊥ =2 fluctuations leads to logarithmically singular renormalizations of the quasiparticle residue Z and the vertex Γ. We derive and solve renormalization-group equations for the flow of Z and Γ, and show that the system develops an anomalous dimension at the nematic quantum critical point (QCP). As a result, the spectral function at a fixed ω and varying k has a non-Lorentzian form. Away from the QCP, we find that the flow of Z is cut at the energy scale ωFL ∝ ξ^-1, associated with the z=1 dynamics of electrons. The z⊥ =2 energy scale, ω∼ ξ^-2, affects the flow of Z only if one includes into the theory self-interaction of transverse fluctuations.