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

We study the dynamic response of a two-dimensional system of itinerant fermions in the vicinity of a uniform ($\mathbf{Q}=0$) Ising nematic quantum critical point of $d-$wave symmetry. The nematic order parameter is not a conserved quantity, and this permits a nonzero value of the fermionic polarization in the $d-$wave channel even for vanishing momentum and finite frequency: $\mathrm{\Pi }(\mathbf{q}=0,{\mathrm{\Omega }}_m)\mathrm{\ne }0$. 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 $\Pi ($\mathbf{q}$ = 0,\Omega_m) \neq 0$ over a parametrically broad range of frequencies where the fermionic self-energy $\mathrm{\Sigma }(\omega )$ is irrelevant, and use Eliashberg theory to compute $\mathrm{\Pi }(\mathbf{q}=0,{\mathrm{\Omega }}_m)$ in the non-Fermi liquid regime at smaller frequencies, where $\mathrm{\Sigma }(\omega )>\omega$. We find that $\mathrm{\Pi }(\mathbf{q}=0,\mathrm{\Omega })$ is a constant, plus a frequency dependent correction that goes as $\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }$ at high frequencies, crossing over to $\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{1\mathrm{/}3}$ at lower frequencies. The $\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{1\mathrm{/}3}$ scaling holds also in a non-Fermi liquid regime. The non-vanishing of $\mathrm{\Pi }(\mathbf{q}=0,\mathrm{\Omega })$ gives rise to additional structure in the imaginary part of the nematic susceptibility ${\chi }^{{}^{\mathrm{\prime }\mathrm{\prime }}}(\mathbf{q},\mathrm{\Omega })$ at $\mathrm{\Omega }>v_{F}q$, 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 $d-$wave geometry.