Dynamical magnetic susceptibility in the spin-fermion model for cuprate superconductors

Using the method of diagram techniques for the spin and Fermi operators in the framework of the $SU(2)$-invariant spin-fermion model of the electron structure of the CuO$_2$ plane of copper oxides, we obtain an exact representation of the Matsubara Green's function $D_\perp(k,i\omega_m)$ of the subsystem of localized spins. This representation includes the Larkin mass operator $\Sigma_{\mathrm L}(k,i\omega_m)$ and the strength and polarization operators $P(k,i\omega_m)$ and $\Pi(k,i\omega_m)$. The calculation in the one-loop approximation of the mass and strength operators for the Heisenberg spin system in the quantum spin-liquid state allows writing the Green's function $D_\perp(k,i\omega_m)$ explicitly and establishing a relation to the result of Shimahara and Takada. An essential point in the developed approach is taking the spin-polaron nature of the Fermi quasiparticles in the spin-fermion model into account in finding the contribution of oxygen holes to the spin response in terms of the polarization operator $\Pi(k,i\omega_m)$.