Fermi liquid near Pomeranchuk quantum criticality

Abstract

We analyze the behavior of an itinerant two-dimensional Fermi system near a charge nematic $(n=2)$ Pomeranchuk instability in terms of the Landau Fermi-liquid (FL) theory. A key object of our study is the fully renormalized vertex function ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$, related to the Landau interaction function. We derive ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$ 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 ${Z}_{\ensuremath{\Gamma}}$, related to the single-particle residue $Z$ and effective-mass renormalization ${m}^{\ensuremath{\ast}}/m$. We employ the Pitaevski-Landau identities, which express the derivatives of the self-energy in terms of ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$, to obtain and solve a set of coupled nonlinear equations for ${Z}_{\ensuremath{\Gamma}}$, $Z$, and ${m}^{\ensuremath{\ast}}/m$. We show that near the transition the system enters a critical FL regime, where ${Z}_{\ensuremath{\Gamma}}\ensuremath{\sim}Z\ensuremath{\propto}{(1+{g}_{c,2})}^{1/2}$ and ${m}^{\ensuremath{\ast}}/m\ensuremath{\approx}1/Z$, where ${g}_{c,2}$ is the $n=2$ charge Landau component which approaches $\ensuremath{-}1$ 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 $\ensuremath{\Sigma}(K)\ensuremath{\propto}\ensuremath{\int}dPG(P)D(K\ensuremath{-}P)$ is asymptotically exact if one identifies the effective interaction $D$ with the RPA form of ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$.