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We show that the fermionic and bosonic spectrum of $d=2$ fermions at finitedensity coupled to a critical boson can be determined non-perturbatively in thecombined limit $k_F\rightarrow {\infty}$, $N_f \rightarrow 0$ with $N_fk_F$fixed. In this double scaling limit, the boson two-point function is corrected,but only at one-loop. This double scaling limit therefore incorporates theleading effect of Landau damping. The fermion two-point function is determinedanalytically in real space and numerically in (Euclidean) momentum space. Theresulting spectrum is discontinuously connected to the quenched $N_f\rightarrow0$ result. For $\omega \rightarrow 0$ with $k$ fixed the spectrum exhibits thedistinct non-Fermi-liquid behavior previously surmised from the RPAapproximation. However, the exact answer obtained here shows that the RPAresult does not fully capture the IR of the theory.

Two-point function of a $d=2$ quantum critical metal in the limit $k_F\rightarrow\infty$, $N_f\rightarrow 0$ with $N_fk_F$ fixed