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Fermi liquid near Pomeranchuk quantum criticality

We analyze the behavior of an itinerant Fermi system near a charge nematic(n=2) Pomeranchuk instability in terms of the Landau Fermi liquid (FL) theory. The main object of our study is the fully renormalized vertex function $ΓΩ$, related to the Landau interaction function. We derive $Γ^Ω$ for a model case of the long-range interaction in the nematic channel. Already within the Random Phase Approximation (RPA), the vertex is singular near the instability. The full vertex, obtained by resumming the ladder series composed of the RPA vertices, differs from the RPA result by a multiplicative renormalization factor $Z_Γ$, related to the single-particle residue $Z$ and effective mass renormalization $m^*/m$. We employ the Pitaevski-Landau identities, which express the derivatives of the self-energy in terms of $Γ^Ω$, to obtain and solve a set of coupled non-linear equations for $Z_Γ$, $Z$, and $m^*/m$. We show that near the transition the system enters a critical FL regime, where $Z_Γ\sim Z \propto (1 + g_{c,2})^{1/2}$ and $m^*/m \approx 1/Z$, where $g_{c,2}$ is the $n=2$ charge Landau component which approaches -1 at the instability. We construct the Landau function of the critical FL and show that all but $g_{c,2}$ Landau components diverge at the critical point. We also show that in the critical regime the one-loop result for the self-energy $Σ(K) \propto \int dP G(P) D (K-P)$ is asymptotically exact if one identifies the effective interaction $D$ with the RPA form of $Γ^Ω$.