  Introducing both Berry curvature and chiral anomaly into Landau's
Fermi-liquid theory, we construct a topological Fermi-liquid theory, applicable
to interacting Weyl metals in the absence of time reversal symmetry. Following
the Landau's Fermi-liquid theory, we obtain an effective free-energy functional
in terms of the density field of chiral fermions. The density field of chiral
fermions is determined by a self-consistent equation, minimizing the effective
free-energy functional with respect to the order-parameter field. Beyond these
thermodynamic properties, we construct Boltzmann transport theory to encode
both the Berry curvature and the chiral anomaly in the presence of forward
scattering of a Fermi-liquid state, essential for understanding dynamic
correlations in interacting Weyl metals. This generalizes the Boltzmann
transport theory for the Landau's Fermi-liquid state in the respect of
incorporating the topological structure and extends that for noninteracting
Weyl metals in the sense of introducing the forward scattering. Finally, we
justify this topological Fermi-liquid theory, generalizing the
first-quantization description for noninteracting Weyl metals into the
second-quantization representation for interacting Weyl metals. First, we
derive a topological Fermi-gas theory, integrating over high-energy electronic
degrees of freedom deep inside a pair of chiral Fermi surfaces. As a result, we
reproduce a topological Drude model with both the Berry curvature and the
chiral anomaly. Second, we take into account interactions between such
low-energy chiral fermions on the pair of chiral Fermi surfaces. We perform the
renormalization group analysis, and find that only forward scattering turns out
to be marginal above possible superconducting transition temperatures,
justifying the topological Fermi-liquid theory of interacting Weyl metals with
time reversal symmetry breaking.
