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(parallel to)=3 and z(perpendicular to)=2, respectively. Previous studies have found that such a system exhibits multiscale quantum criticality with two different energy scales omega similar to xi(-z parallel to,perpendicular to), where xi 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(parallel to)=3 mode is known to give rise to a local self-energy that develops a non-Fermi-liquid form, Sigma(omega)similar to omega(2/3) for frequencies larger than the energy scale omega similar to xi(-3). We find that the exchange of transverse z(perpendicular to)=2 fluctuations leads to logarithmically singular renormalizations of the quasiparticle residue Z and the vertex Gamma. We derive and solve renormalization-group equations for the flow of Z and Gamma, 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 omega 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 omega(FL)proportional to xi(-1), associated with the z=1 dynamics of electrons. The z(perpendicular to)=2 energy scale, omega similar to xi(-2), affects the flow of Z only if one includes into the theory self-interaction of transverse fluctuations.