Infinite-range limit for correlation functions of lattice systems

For a lattice Fermi gas, the quantum and classical Heisenberg models, and the Ising model it is shown that in the limit of an interaction of infinite range the correlation functions of these systems are identical to the expressions for them obtained in the self-consistent field approximation. The Lebowitz–Penrose theorem is also proved by a modified method of N. N. Bogolyubov (Jr). It is shown in the Appendix that the number of interacting harmonics in the method of the approximating Hamiltonian admits any growth less than the growth of the volume of the system.