A unified description of static and dynamic properties of Fermi liquids

Abstract

In Landau's phenomenological Fermi-liquid theory (FLT), most physical quantities are derived from the knowledge of the energy variation $\delta E[\delta n]$ corresponding to a change $\delta n$ of the quasi-particle (QP) distribution function $n \equiv {n_{k \sigma}}$. We show that the internal energy $E[n]$ (or, more precisely, the thermodynamic potential $\Phi[n]$), expressed as a function of the QP distribution $n$, can be interpreted as an effective potential (in the sense of field theory), which is obtained from the free energy by a Legendre transformation. This allows to obtain explicitly $\delta\Phi$ (or $\delta E$) starting from a microscopic Hamiltonian and to relate the Landau $f$ function to the forward-scattering two-particle vertex without considering the collective modes as in the standard diagrammatic derivation of FLT. Out-of-equilibrium properties are obtained by extending the definition of the effective potential to space- and time-dependent configurations. $\Phi[n]$ is then a functional of the Wigner distribution function $n \equiv {n_{k \sigma}(r,t)}$. It contains information about both the static and dynamic properties of the Fermi liquid. In particular, it yields the quantum Boltzmann equation satisfied by $n_{k \sigma}(r,t)$. Finally, we show how $\delta\Phi[\delta n]$ can be derived (in the static case) using a finite-temperature renormalization-group approach. In agreement with previous results based on this technique, we find that the Landau $f$ function is defined by the fixed-point value of the $\Omega$-limit of the forward-scattering two-particle vertex.