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.

中文翻译：

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二维费米液体中集体模式的非典型行为

利用朗道动力学方程研究相互作用的费米系统的非平衡行为是朗道费米液体理论的最高成就之一。虽然已经对标准的三维费米液体的传输模式进行了彻底的研究，但缺乏对二维费米液体的同样深入的分析。在将朗道动力学方程 (LKE) 应用于二维费米液体时，我们获得了零声模式 $c_0$ 的非常规行为。作为通常的无量纲参数 $s=\omega/qv_F$ 的函数，我们发现了两个特殊的结果：首先，对于 $|s|>1$，我们看到弱相互作用系统的无阻尼模式的传播。这与三维情况不同，在三维情况下，无阻尼模式仅在排斥相互作用中传播，而模式在任何任意吸引相互作用时都会经历朗道阻尼。其次，我们发现无论交互强度如何，$|s|<1$ 都禁止传播模式。这与模式可以传播（尽管受到阻尼）的三维情况截然不同。此外，我们提出了二维费米液体的修正 Pomeranchuk 不稳定性条件以及直接来自 LKE 的流体的运动方程。在二维中，我们为 $\ell\geq 1$ 的所有朗道参数找到了一个常数最小值，这与三维情况不同。