Soft modes in Fermi liquids at arbitrary temperatures

Abstract

We use kinetic-theory methods to analyze Landau Fermi-liquid theory, and in particular to investigate the number and nature of modes in Fermi liquids that are soft in the long-wavelength limit, both in the hydrodynamic and the collisionless regimes. In the hydrodynamic regime we show that Fermi-liquid theory is consistent with Navier-Stokes hydrodynamics at all temperatures, as expected. The soft modes are the ones familiar from classical hydrodynamics that are controlled by the five conservation laws; namely, two first-sound modes, two shear diffusion modes, and one heat diffusion mode. These modes have a particle-like spectrum and are soft, or scale invariant, at all temperatures. In the collisionless regime we show that the entire single-particle distribution function is soft with a continuous part of the spectrum. This continuous soft mode, which is well known but often not emphasized, has important physical consequences, e.g., for certain quantum phase transitions. In addition, there are the well-known soft zero-sound excitations that describe angular fluctuations of the Fermi surface; their spectra are particle-like. They are unrelated to conservation laws, acquire a mass at any nonzero temperature, and their number depends on the strength of the quasiparticle interaction. We also discuss the fates of these two families of soft modes as the temperature changes. With increasing temperature the size of the collisionless regime shrinks, the damping of the modes grows, and eventually all of the collisionless modes become overdamped. In their stead the five hydrodynamic modes appear in the hydrodynamic regime at asymptotically low frequencies. The two families of soft modes are unrelated and have very different physical origins. In charged Fermi liquids the first-sound modes in the hydrodynamic regime and the $\ensuremath{\ell}=0$ zero-sound modes in the collisionless regime get replaced by plasmons, all other modes remain soft.