Limits of the exchange-correlation local fields in the magnetic response of a spin-polarized electron gas

We analyze the spin and charge susceptibilities of a spin-polarized electron gas subject to a weak space- and time-dependent field coupled to the electronic spins, with the main attention given to the case of space dimensionality D = 2. Exchange and correlations enter the dynamic susceptibilities through spin-dependent local-field factors G(sigma)(+/-)(q,omega). For an arbitrary degree of polarization, we determine the exact analytic expressions of G(sigma)(+/-)(q,omega) in two limiting cases: (i) the limit of large wave number q at finite frequency w, already considered in D = 3 by D. C. Marinescu and J. J. Quinn [Phys. Rev. B 56, 1114 (1997)]; and (ii) the static limit at small wave number. In the latter case, we obtain thermodynamic sum rules of general validity in both dimensionalities. Our work fives insight into many-body vertex corrections and basic information for calculations of the effects of the electron-electron interactions on physical properties.