Fermi liquid near Pomeranchuk quantum criticality

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

Figure 1

Fermi-liquid vertices to second order in the bare interaction (wavy line). (a) Diagrams for ${\Gamma }_{\alpha \beta ;\gamma \delta }^{\Omega }$. (b) Diagrams for ${\Gamma }_{\alpha \beta ;\gamma \delta }^q$. (c) Diagrammatic representation of the relation between ${\Gamma }_{\alpha \beta ;\gamma \delta }^{\Omega }$ and ${\Gamma }_{\alpha \beta ;\gamma \delta }^q$, Eq. (2.3). A bubble composed of dotted lines represents the particle-hole propagator $\mathcal{P}$ [see Eq. (2.2)] in the limit $\Omega /q\to 0$, where $\mathcal{P}$ reduces to a constant.

Figure 2

RPA series for the nematic susceptibility [(a)] and ${\Gamma }_{\alpha \beta ;\gamma \delta }^q$ [(b)]. Solid circles represent $d$-wave vertices $d_{\mathbf{k}}$ and wavy lines represent the forward-scattering part of the interaction, $U_0$. The bubbles are with zero vanishing external momentum and zero frequency, the same bubbles as in Figs. 1b, 1c. Since, by assumption $a{k}_{\text{F}}⪢1$, only the diagrams that contain $U_0$ are included.

Figure 3

RPA diagrams for ${\Gamma }_{\alpha \beta ;\gamma \delta }^{\Omega }$. The difference with the series for ${\Gamma }_{\alpha \beta ;\gamma \delta }^q$ in Fig. 2b is in that the polarization bubbles are evaluated at finite external momentum $\mathbf{k}-\mathbf{p}$.

Figure 4

(a) Ladder series for the full vertex ${\Gamma }_{\alpha \beta ;\gamma \delta }^{\Omega }$ beyond RPA. Each hatched block represents ${\Gamma }_{\alpha \beta ;\gamma \delta }^{\Omega }$ at the RPA level, Eq. (3.23). (b) One-loop vertex correction to the ladder diagram.

Figure 5

Top: schematic behavior of the charge $\left(c\right)$ and spin $\left(s\right)$ components of the critical Landau function $\overline{g}$ as a function of $g_{c,2}$. Bottom: same for the actual Landau function $g^{\ast }$.

Figure 6

(a) Third-order diagrams for ${\Gamma }_{\alpha \beta ;\gamma \delta }^{\Omega }$ that contain only the particle-particle and particle-hole bubbles at combinations of external momenta $K\pm P$. (b) Examples of the third-order diagrams that contain convolutions of particle-particle and particle-hole bubbles with Green’s functions over the internal momenta.

Figure 7

Diagrams for the self-energy at the one-loop [(a)] and two-loop orders [(b) and (c)]. The wavy line is the RPA interaction, Eq. (3.23).