Singular three-point density correlations in two-dimensional Fermi liquids

Abstract

We characterize a singularity in the equal-time three-point density correlations that is generic to two-dimensional interacting Fermi liquids. In momentum space where the three-point correlation is determined by two wavevectors $\mathbf{q}_1$ and $\mathbf{q}_2$, the singularity takes the form $|\mathbf{q}_1\times\mathbf{q}_2|$. We explain how this singularity is sharply defined in a long-wavelength collinear limit. For a non-interacting Fermi gas, the coefficient of this singularity is given by the quantized Euler characteristic of the Fermi sea, and it implies a long-range real space correlation favoring collinear configurations. We show that this singularity persists in interacting Fermi liquids, and express the renormalization of the coefficient of singularity in terms of Landau parameters, for both spinless and spinful Fermi liquids. Implications for quantum gas experiments are discussed.

Figures (4)

• Figure 1: (a) Momentum space triangle formed by $\{{\bf q}_{1,2,3}\}$. (b) $s_3$ as a function of $q_{\perp}$ in the LWC limit based on Eq. \ref{['eq: collinear formula for cFS']}. The $\abs{q_{\perp}}$ singularity in Eq. \ref{['eq: topological formula, momentum space']} (dashed line) is smoothened only for $\abs{q_{\perp}}\lesssim \mathcal{O}(\abs{{\bf q}}^2/k_F)$. (c) In the LWC limit, $s_3$ is dominated by region near the Fermi surface critical points where ${\bf v} \perp {\bf q}_{a}$. The integrand $\mathcal{S}({\bf k}; {\bf q}_1, {\bf q}_2)=0$ and $1$, respectively, for situation in (i) and (ii). (d) Long-range collinear correlation in real space, which corresponds to the momentum space triangle in (a). The collinear direction in real space is orthogonal to the collinear direction in momentum space.
• Figure 2: (a) Three-point bubble $\Pi_3$ dressed with vertex renormalizations. (b) Polarization bubble $\Pi^0_2$. (c) Dyson equation for the vertex renormalization $\Lambda$ in spinless Fermi liquids. (d) Vertex renormalizations, $\Lambda^{\uparrow\uparrow}$ and $\Lambda^{\uparrow\downarrow}$, in spinful Fermi liquids. Vertices (labeled as dots) represent electron density operators, solid lines represent fermionic propagators, and dashed lines represent renormalized density-density forward-scattering interactions.
• Figure 2.1: Feynman diagrams for calculating the three-point density correlation in a Fermi liquid. (a) The three-point bubble $\Pi_3$ is dressed with vertex renormalizations (indicated as shaded vertices) that encode interaction effects. $\Pi_3$ (as well as its non-interacting version $\Pi^0_3$) is contributed by two diagrams and correspond respectively to the two terms in Eq. \ref{['supp_eq: Pi0_3 from diagrams']} and Eq. \ref{['supp_eq: Pi_3 from diagrams']}. (b) The vertex renormalization $\Lambda_{\bf k}({\bf q},\tau'-\tau)$ obeys a Dyson equation, see Eq. \ref{['supp_eq: Dyson equation in time domain']}, where interaction effects are accounted for by summing a geometric series of polarization bubble diagrams. $V_{{\bf k}'{\bf k}}$ is the effective forward scattering interaction and represented as a dashed line.
• Figure 3.1: (a) Feynman diagram for the contribution to the three-point density correlation from a three-body density-density interaction $U_{\{{\bf k}_a\}}$. (b) Dependence of $I$, defined in Eq. \ref{['supp_eq: delta s_3 from 3-body']}, on the angle $\varphi_{12}$ between ${\bf q}_1$ and ${\bf q}_2$, showing that the three-body interaction does not generate $\abs{{\bf q}_1\times{\bf q}_2}$-type correction to $s_3$.