Gradient expansion approach to multiple-band Fermi liquids

Promoted by the recent progress of Berry phase physics in spin galvanomagnetic communities, we develop a systematic derivation of the reduced Keldysh equation (RKE) which captures the low-energy dynamics of quasiparticles constrained within doubly degenerate bands forming a single Fermi surface. The derivation begins with the Keldysh equation for a quite general multiple-band interacting Fermi system, which is originally an N(b)xN(b) matrix-formed integral (or infinite-order differential) equation, with N-b being the total number of bands. To derive the RKE for quasiparticle on a Fermi surface in question, we project out both the fully occupied and empty band degrees of freedom perturbatively in the gradient expansion, whose coupling constant measures how a system is disequilibrated. As for the electron-electron interactions, however, we only employ the so-called adiabatic assumption of the Fermi liquid theory, so that the electron correlation effects onto the adiabatic transport of quasiparticles, i.e., the Hermitian (real) part of the self-energy, are taken into account in an unbiased manner. The RKE thus derived becomes an SU(2) covariant differential equation and treats the spin and charge degrees of freedom on an equal footing. Namely, the quasiparticle spin precessions due to the non-Abelian gauge fields are automatically encoded into its covariant derivatives. When further solved in favor of spectral functions, this covariant differential equation suggests that quasiparticles on a doubly degenerate Fermi surface acquire spin-selective Berry curvature corrections under the applied electromagnetic fields. This theoretical observation gives us some hints of possible experimental methodology for measuring the SU(2) Berry's curvatures by spin-resolved photoemission experiments. Due to the nontrivial frequency dependence of (the Hermitian part of) self-energy, our RKE is composed of Berry's curvatures in the d+1 dual space, i.e., k-omega space, so that the dual electric field is already introduced. To provide a simple way to understand this temporal component of the U(1) Berry's curvature, we also provide the dual analog of the Ampere's law, where the spatial rotation of the electric field in combination with the temporal derivative of the well-known magnetic component is determined by the U(1) magnetic monopole current..