We develop a theoretical framework based on the nonperturbative renormalization group (RG) in the one-particle irreducible (Wetterich) formulation to tackle the interplay of coupled fermionic and order-parameter fluctuations at metallic quantum critical points (QCPs) with ordering wave vector $\vec{Q}=\vec{0}$. We consistently treat the dynamical emergence of the Landau damping of the bosonic mode and non-Fermi liquid scaling of fermions upon lowering the cutoff scale. The loop integrals of the present theory involve only contributions from fluctuations above the cutoff scale, which protects the system from developing singular bosonic interactions. We emphasize the importance of the nontrivial relative scaling of the bosonic and fermionic cutoffs $\mathrm{\Lambda }$ and ${\mathrm{\Lambda }}_f$, which we fix by analyzing the RG flow of the scale-dependent ordering wave-vector ${\vec{Q}}_{\mathrm{\Lambda }}$. Upon neglecting Fermi self-energy in the loop integrals of the functional RG flow, we identify a non-Fermi liquid RG fixed point and recover the features obtained earlier within random phase approximation-type approaches. In a subsequent step, we self-consistently include the scaling of the self-energy and the Yukawa coupling. We find a generic instability of the non-Fermi liquid RG fixed point. This implies, at least at this truncation level, absence of the QCP with $\vec{Q}=\vec{0}$ and development of a first-order phase transition or a phase characterized by $\vec{Q}\ne \vec{0}$.

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• Fermions
• Quantum phase transitions
• 2-dimensional systems
• Metals
• Quantum many-body systems
• Strongly correlated systems
• Fermi liquid theory
• Non-Fermi-liquid theory
• Renormalization group