Collective effects in an incompressible electronic liquid

Starting from the Landau's kinetic equation, we show that an electronic liquid in $d=2,3$ dimensions depicted by a Landau type effective theory will become incompressible on condition that the Landau parameters satisfy either (i) $1+F_{1}^{s}/d=0$ or (ii) $F_{0}^{s}\to{}+\infty$. The condition (i) is the Pomeranchuk instability in the current channel and suggests a quantum spin liquid (QSL) state with a spinon Fermi surface; while the condition (ii) means that the strong repulsion in the charge channel and leads to a conventional charge and thermal insulator. In the collisionless regime ($\omega\tau\gg{}1$) and the hydrodynamic regime ($\omega\tau\ll{}1$), the zero and first sound modes have been studied and classified by symmetries, including the longitudinal and transverse modes in $d=2,3$ and the higher angular momentum modes in $d=3$. The sufficient (and/or necessary) conditions of these collective modes have been revealed. It has been demonstrated that some of these collective modes will behave in quite different manners under two incompressibility conditions (i) or (ii). Possible nematic QSL states and a hierarchy structure for gapless QSL states have been proposed in $d=3$.