Collective effects in an incompressible electronic liquid

Starting from Landau's kinetic equation, we show that an electronic liquid in d = 2, 3 spatial dimensions depicted by a Landau-type effective theory will become incompressible on condition that the Landau parameters satisfy either (i) [Formula: see text] or (ii) [Formula: see text]. Condition (i) is the Pomeranchuk instability in the current channel and suggests a quantum spin liquid (QSL) state with a spinon Fermi surface; while condition (ii) means that the strong repulsion in the charge channel leads to a conventional charge and thermal insulator. In the collisionless regime (ωτ ≫ 1) and the hydrodynamic regime (ωτ ≪ 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 incompressibility condition (i) or (ii). Possible nematic QSL states and a hierarchy structure for gapless QSL states have been proposed in d = 3.