  Collective electronic fluctuations in correlated materials give rise to
various important phenomena, such as existence of the charge ordering,
superconductivity, Mott insulating and magnetic phases, plasmon and magnon
modes, and other interesting features of such systems. Unfortunately,
description of these correlation effects requires significant efforts, since
they almost entirely rely on strong local and nonlocal electron-electron
interactions. Some collective phenomena, such as magnetism, can be sufficiently
described by a simple Heisenberg-like models that are formulated in terms of
bosonic variables. This fact suggests that other many-body excitations can also
be described by simple bosonic models in spirit of the Heisenberg theory. Here
we derive an effective bosonic action for charge degrees of freedom for the
extended Hubbard model and define a physical regime where the obtained action
reduces to a classical Hamiltonian of an effective Ising model.
