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: $\Pi(\mathbf{ q} = 0,\Omega_m) \neq 0$. 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 $\Sigma (\omega)$ is irrelevant, and use Eliashberg theory to compute $\Pi (\mathbf{ q} = 0,\Omega_m)$ in the non-Fermi liquid regime at smaller frequencies, where $\Sigma (\omega) > \omega$. We find that $\Pi(\mathbf{ q}=0,\Omega)$ is a constant, plus a frequency dependent correction that goes as $|\Omega|$ at high frequencies, crossing over to $|\Omega|^{ 1/3}$ at lower frequencies. The $|\Omega|^{ 1/3}$ scaling holds also in a non-Fermi liquid regime. The non-vanishing of $\Pi (\mathbf{ q}=0, \Omega)$ gives rise to additional structure in the imaginary part of the nematic susceptibility $\chi^{ ''} (\mathbf{ q}, \Omega)$ at $\Omega > v_F q$, 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.
Categories: cond-mat.str-el cond-mat.supr-con