A unified description of static and dynamic properties of Fermi liquids

TL;DR

This paper presents a unified framework for describing both static and dynamic properties of Fermi liquids by interpreting the thermodynamic potential as an effective potential, linking microscopic Hamiltonians to Landau parameters, and extending to out-of-equilibrium states.

Contribution

It introduces a field-theoretic interpretation of the thermodynamic potential in Fermi-liquid theory and extends the formalism to out-of-equilibrium and dynamic properties.

Findings

Relates the thermodynamic potential to an effective potential via Legendre transformation.

Derives the Landau f function from the microscopic two-particle vertex.

Extends the formalism to space- and time-dependent out-of-equilibrium configurations.

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.