One-dimensional scattering of two-dimensional fermions near quantum criticality

TL;DR

This paper investigates the scattering rates of two-dimensional fermions near quantum criticality, revealing the importance of planar processes and their impact on scattering behavior, especially near a nematic transition and in superconducting states.

Contribution

It extends Fermi liquid theory to include higher-order logarithmic corrections from planar processes and sums these leading logs near a nematic transition, providing a comprehensive analysis of scattering rates.

Findings

Higher powers of log(ω) appear in backscattering at higher orders.

Leading logarithms are summed for a 2D Fermi liquid near a nematic transition.

Scattering rate is suppressed for repulsive interactions and increases near the superconducting gap for attractive interactions.

Abstract

Forward and backscattering play an exceptional role in the physics of two-dimensional interacting fermions. In a Fermi liquid, both give rise to a non-analytic $\omega^2 \log(\omega)$ form of the fermionic scattering rate at second order in the interaction. Here we argue that higher powers of $\log(\omega)$ appearin the backscattering contribution at higher orders. We show that these terms come from "planar" processes, which are effectively one-dimensional. This is explicitly demonstated by extending a Fermi liquid to the limit of $N \gg 1$ fermionic flavors, when only planar processes survive. We sum the leading logarithms for the case of a 2D Fermi liquid near a nematic transition and obtain an expression for the scattering rate at $T=0$ to all orders in the interaction. For a repulsive interaction, the resulting rate is logarithmically suppressed, and the result is valid down to $\omega = 0$. For an attractive interaction, the ground state is an $s$-wave superconductor with a gap $\Delta_0$. We show that in this case the scattering rate increases as $\omega$ is reduced towards $\Delta_0$. At $\omega \geq \Delta_0$, the behavior of the scattering rate is rather unconventional as many pairing channels compete near a nematic critical point, and $s$-wave wins only by a narrow margin. We take superconductivity into consideration and obtain the scattering rate also at smaller $\omega \simeq \Delta_0$.