  We review a representation of Hubbard-like models that is based on auxiliary
pseudospin variables. These pseudospins refer to the local charge modulo two in
the original model and display a local Z_2 gauge freedom. We discuss the
associated mean-field theory in a variety of different contexts which are
related to the problem of the interaction-driven metal-insulator transition at
half-filling including Fermi surface deformation and spectral features beyond
the local approximation. Notably, on the mean-field level, the Hubbard bands
are derived from the excitations of an Ising model in a transverse field and
the quantum critical point of this model is identified with the Brinkman-Rice
criticality of the almost localized Fermi liquid state. Non-local correlations
are included using a cluster mean-field approximation and the Schwinger boson
theory for the auxiliary quantum Ising model.
