Quasiclassical theory of spin dynamics in superfluid $^3$He: kinetic equations in the bulk and spin response of surface Majorana states

TL;DR

This paper develops a quasiclassical Green's function framework to analyze spin dynamics in superfluid $^3$He, including bulk behavior and surface Majorana states, with a focus on spin relaxation and Fermi-liquid effects.

Contribution

It introduces a comprehensive quasiclassical kinetic theory for spin dynamics in superfluid $^3$He, extending previous results and incorporating surface effects and Fermi-liquid corrections.

Findings

Derived kinetic equations for bulk superfluid spin dynamics.

Analyzed spin relaxation near superfluid surfaces with localized fermionic states.

Calculated spin wave lifetimes considering Fermi-liquid modifications.

Abstract

We develop a theory based on the formalism of quasiclassical Green's functions to study the spin dynamics in superfluid $^3$He. First, we derive kinetic equations for the spin-dependent distribution function in the bulk superfluid reproducing the results obtained earlier without quasiclassical approximation. Then we consider a spin dynamics near the surface of fully gapped $^3$He-B phase taking into account spin relaxation due to the transitions in the spectrum of localized fermionic states. The lifetime of longitudinal and transverse spin waves is calculate taking into account the Fermi-liquid corrections which lead to the crucial modification of fermionic spectrum and spin responses.