Atypical Behavior of Collective Modes in Two-Dimensional Fermi Liquids

TL;DR

This paper investigates the unique behavior of collective modes in two-dimensional Fermi liquids using the Landau kinetic equation, revealing unconventional propagation characteristics of zero sound modes and revising stability criteria specific to 2D systems.

Contribution

It provides the first detailed analysis of collective modes in 2D Fermi liquids, highlighting differences from 3D systems and introducing revised instability conditions and equations of motion.

Findings

Undamped zero sound mode propagates for |s|>1 in weakly interacting 2D Fermi liquids.

Propagating modes are forbidden for |s|<1 regardless of interaction strength.

Coulomb interactions cause a crossover from zero sound to plasmon modes.

Abstract

Using the Landau kinetic equation to study the non-equilibrium behavior of interacting Fermi systems is one of the crowning achievements of Landau's Fermi liquid theory. While thorough study of transport modes has been done for standard three-dimensional Fermi liquids, an equally in-depth analysis for two dimensional Fermi liquids is lacking. In applying the Landau kinetic equation (LKE) to a two-dimensional Fermi liquid, we obtain unconventional behavior of the zero sound mode $c_0$. As a function of the usual dimensionless parameter $s=\omega/qv_F$, we find two peculiar results: First, for $|s|>1$ we see the propagation of an undamped mode for weakly interacting systems. This differs from the three dimensional case where an undamped mode only propagates for repulsive interactions and the mode experiences Landau damping for any arbitrary attractive interaction. Second, we find that regardless of interaction strength, a propagating mode is forbidden for $|s|<1$. This is profoundly different from the three-dimensional case where a mode can propagate, albeit damped. In addition, we present a revised Pomeranchuk instability condition for a two-dimensional Fermi liquid as well as equations of motion for the fluid that follow directly from the LKE. In two dimensions, we find a constant minimum for all Landau parameters for $\ell\geq 1$ which differs from the three dimensional case. Finally we discuss the effect of a Coulomb interaction on the system resulting in the plasmon frequency $\omega_p$ exhibiting a crossover to the zero sound mode.