Instability of the critical Ngai's coupling and two-boson mechanism in metals

We study the properties of a Fermi liquid coupled to a quantum critical boson via the two-boson interaction known as Ngai's coupling. We find that the original quantum critical point is generally unstable, resulting in a finite-momentum spatially modulated state unless two conditions are satisfied: (i) the critical boson is polar and transverse, and (ii) the ratio of the Fermi velocity to the transverse-boson velocity is sufficiently large. If these conditions hold and the uniform state remains stable, we demonstrate that the system enters a strong-coupling regime below a certain energy scale. In this regime, we discuss a self-consistent solution at criticality in two-dimensional systems and show that the critical boson field develops a nontrivial anomalous dimension, $\eta=1/2$. Our findings highlight the significant role of two-boson coupling in critical theories, challenging the conventional view that its effects are subdominant to linear coupling.