Whenever mean-field approximations (MFA) can be made to a Hamiltonian, they not only simplify significantly its solution, but in most cases, also provide new physical insight. The simplest MFA to a Hamiltonian of interacting electrons is the Hartree-Fock or SCF approximation. It neglects totally electron correlations. The question arises whether MFAs can be made which take strong correlations reasonably well into account. Of special interest are clearly mean-field solutions with broken symmetries, i.e., with a ground state which has a lower symmetry than the Hamiltonian. However, care must be exercised here. Often only a symmetry breaking occurs, because correlation effects are simulated which otherwise are insufficiently taken into account.
For example, an AF ground state reduces on-site charge fluctuations as do correlations. In that case, one has to decide whether an AF mean-field solution describes the system correctly or whether it is merely favored because it suppresses charge fluctuations. The simplest example is a H2 molecule with variable bond length. An unrestricted SCF calculation gives a symmetry broken ground state when the bond length exceeds a critical value. Near one proton the electrons have predominantly spin up while near the other proton they have predominantly spin down. In reality, there is, of course, no distinction between the spins on the two sites. But therefore, all ionic configurations are eliminated by the unrestricted MFA in the limit of large bond lengths.
A special object of mean-field investigations has been the Hubbard model on a cubic or square lattice at or near half filling. With hopping limited to nearest neighbors one finds for the half-filled case perfect nesting at the Fermi surface and a spin-density mean-field ground state for any value of U >0.
An interesting variety of MF solutions is obtained when auxiliary slave boson or fermion fields are introduced  (see Chap. 3). An example is the replacement at- = f-b+, a+ = f+bt, (1.13)
where the fermion operator f++ creates a spinon at site t while the boson bt+ creates an empty site (holon). Strong correlations can be taken into account by forbidding double occupancies via the subsidiary condition
i.e., a site is either empty or singly occupied. When the t — J Hamiltonian (1.10) is re-expressed in terms of spinons and holons, different MFAs with different order parameters can be made . One of them has the form of a BCS superconducting order parameter, but here in terms of spinons, not electrons, while another is of resonating valence bond (RVB) type (see Chap. 1.6). There are also other MFAs possible, in particular, since the decomposition (1.14) of electrons in form of spinons and holons is not the only possible one. Another decomposition is a+ = fb+, at- = f+bt-, (1.15)
where the spin degree of freedom is represented by a boson field. We want to find a representation of the Hamiltonian in terms of auxiliary fields in terms of which a MFA describes strong electron correlations as well as possible.
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