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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 polarizations, longitudinal and transverse to the quadrupolar momentum tensor. Longitudinal and transverse bosonic modes are characterized by dynamical exponents z_parallel=3 and z_perp=2, respectively. Previous studies have found that such a system exhibits multiscale quantum criticality with two different energy scales omega ~ xi^{-z_{parallel,perp}}, where xi is the correlation length. We study the impact of the multiple energy scales on the electron Green function. The interaction with the critical z_parallel =3 mode is known to give rise to a local self-energy that develops a non-Fermi liquid form, Sigma(omega) ~ omega^{2/3} for frequencies larger than the energy scale omega ~ xi^{-3}. We find that the exchange of transverse z_perp=2 fluctuations leads to a 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} ~ xi^{-1}, associated with the z=1 dynamics of electrons. The z_perp=2 energy scale, omega ~ xi^{-2}, affects the flow of Z only if one includes into the theory self-interaction of transverse fluctuations.