Dynamical susceptibility near a long-wavelength critical point with a nonconserved order parameter

Abstract

We study the dynamic response of a two-dimensional system of itinerant fermions in the vicinity of a uniform $(\mathrm{Q}=0)$ Ising nematic quantum critical point of $d\text{\ensuremath{-}}\text{wave}$ symmetry. The nematic order parameter is not a conserved quantity, and this permits a nonzero value of the fermionic polarization in the $d\text{\ensuremath{-}}\text{wave}$ channel even for vanishing momentum and finite frequency: $\mathrm{\ensuremath{\Pi}}(\mathrm{q}=0,{\mathrm{\ensuremath{\Omega}}}_{m})\ensuremath{\ne}0$. For weak coupling between the fermions and the nematic order parameter (i.e., the coupling is small compared to the Fermi energy), we perturbatively compute $\mathrm{\ensuremath{\Pi}}(\mathrm{q}=0,{\mathrm{\ensuremath{\Omega}}}_{m})\ensuremath{\ne}0$ over a parametrically broad range of frequencies where the fermionic self-energy $\mathrm{\ensuremath{\Sigma}}(\ensuremath{\omega})$ is irrelevant, and use Eliashberg theory to compute $\mathrm{\ensuremath{\Pi}}(\mathrm{q}=0,{\mathrm{\ensuremath{\Omega}}}_{m})$ in the non-Fermi-liquid regime at smaller frequencies, where $\mathrm{\ensuremath{\Sigma}}(\ensuremath{\omega})>\ensuremath{\omega}$. We find that $\mathrm{\ensuremath{\Pi}}(\mathrm{q}=0,\mathrm{\ensuremath{\Omega}})$ is a constant, plus a frequency-dependent correction that goes as $|\mathrm{\ensuremath{\Omega}}|$ at high frequencies, crossing over to ${|\mathrm{\ensuremath{\Omega}}|}^{1/3}$ at lower frequencies. The ${|\mathrm{\ensuremath{\Omega}}|}^{1/3}$ scaling holds also in a non-Fermi-liquid regime. The nonvanishing of $\mathrm{\ensuremath{\Pi}}(q=0,\mathrm{\ensuremath{\Omega}})$ gives rise to additional structure in the imaginary part of the nematic susceptibility ${\ensuremath{\chi}}^{\ensuremath{''}}(q,\mathrm{\ensuremath{\Omega}})$ at $\mathrm{\ensuremath{\Omega}}>{v}_{F}q$, in marked contrast to the behavior of the susceptibility for a conserved order parameter. This additional structure may be detected in Raman scattering experiments in the $d\text{\ensuremath{-}}\text{wave}$ geometry.