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 (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: Π(q=0,Ωm)≠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 Π(q=0,Ωm)≠0 over a parametrically broad range of frequencies where the fermionic self-energy Σ(ω) is irrelevant, and use Eliashberg theory to compute Π(q=0,Ωm) in the non-Fermi-liquid regime at smaller frequencies, where Σ(ω)>ω. We find that Π(q=0,Ω) is a constant, plus a frequency-dependent correction that goes as |Ω| at high frequencies, crossing over to |Ω|1/3 at lower frequencies. The |Ω|1/3 scaling holds also in a non-Fermi-liquid regime. The nonvanishing of Π(q=0,Ω) gives rise to additional structure in the imaginary part of the nematic susceptibility χ″(q,Ω) at Ω>vFq, 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.