Long-wavelength behavior of the dynamical spin-resolved local-field factor in a two-dimensional electron liquid

The high-frequency limits of the singular component A(ω) of the small wave-vector expansion of the longitudinal (L) and transverse (T) components of the spin-resolved exchange-correlation kernel tensor fxc,σσ′L,T(q,ω)=−v(q)Gσσ′L,T(q,ω) in a two-dimensional isotropic electron liquid with arbitrary spin polarization are studied. Here Gσσ′L,T(q,ω) is the spin-resolved local-field factor, v(q) is the Coulomb interaction in momentum space, and σ denotes spin. Particularly, the real part of A(ω) is found to be logarithmically divergent at large ω. The large wave-vector structure of the corresponding spin-resolved static structure factor is also established.