  Introducing both the Berry curvature and chiral anomaly into the Landau's
Fermi-liquid theory, we investigate collective dynamics of Fermi-surface
fluctuations and reveal their instabilities in an interacting Weyl metal phase
with broken time reversal symmetry. Based on the Boltzmann-equation framework,
we find criteria for the stability of the topological Fermi-liquid state as a
function of forward scattering Landau's interaction parameters and the distance
of a pair of Weyl points given by an external magnetic field. In addition to
these instability criteria for general angular momentum channels, we
investigate the dispersion relation of the zero-sound mode as the simplest
example of such Fermi-surface fluctuations. Zero sound modes are well-defined
collective excitations in a Landau's Fermi-liquid state, given by the
collective dynamics of Fermi-surface deformations in the spin-singlet channel
with zero angular momentum, where their instability is related with phase
separation. We find that the role of the Berry curvature changes the
instability criteria of the Landau's Fermi-liquid state. Even if the zero-sound
mode is stable in the region of the forward-scattering amplitude, the Berry
curvature gives rise to Landau damping beyond the Landau's Fermi-liquid theory.
Based on the instability criterion of the zero-sound mode, we propose a phase
diagram for a topological Fermi-liquid state against the phase separation in
the plane of Landau's interaction parameter and effective Berry curvature,
which generalizes the one-dimensional phase diagram of the Landau's
Fermi-liquid theory.
