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.
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We thank M. Schuett, R. Fernandes, S. Kivelson, and M. Punk for stimulating discussions. This work was supported by the NSF Grant No. DMR-1523036 (A.K. and A.C.). S.L. and D.C. are supported by a postdoctoral fellowship from the Gordon and Betty Moore Foundation, under the EPiQS initiative, Grant No. GBMF-4303, at MIT.