Bogolyubov's statistical variational principle and the Green's function method applied to the Heisenberg–Ising model

The link between Bogoiyubov's statistical variational principle for free energy, the method of partial diagram summation of the perturbation theory, and the Luttinger–Ward theorem is established. On the basis of Matsubara's Green's function method the nonlinear integral Dyson equation is solved by approximating the effective potential and a new implicit equation of magnetic state is obtained for the Ising model.