Holographic zero sound at finite temperature

We use gauge-gravity duality to study the temperature dependence of the zero sound mode and the fundamental matter diffusion mode in the strongly coupled {\cal N}=4 SU(N_c) supersymmetric Yang-Mills theory with N_f {\cal N}=2 hypermultiplets in the N_c>>1, N_c>>N_f limit, which is holographically realized via the D3/D7 brane system. In the high density limit \mu>>T, three regimes can be identified in the behavior of these modes, analogous to the collisionless quantum, collisionless thermal and hydrodynamic regimes of a Landau Fermi-liquid. The transitions between the three regimes are characterized by the parameters T/\mu and (T/\mu)^2 respectively, and in each of these regimes the modes have a distinctively different temperature and momentum dependence. The collisionless-hydrodynamic transition occurs when the zero sound poles of the density-density correlator in the complex frequency plane collide on the imaginary axis to produce a hydrodynamic diffusion pole. We observe that the properties characteristic of a Landau Fermi-liquid zero sound mode are present in the D3/D7 system despite the atypical T^6/\mu^3 temperature scaling of the specific heat and an apparent lack of a directly identifiable Fermi surface.