While the projection onto the physical Fock space with local charge Q = 1 (3.8) may easily be performed exactly for a single correlated site, it becomes cumbersome already for two sites, let alone for a lattice of correlated electrons. This is because by the exact projection the partition function, for instance, is transformed into the expectation value of the product of all the local charges Qj on the correlated lattice sites j (compare the discussion after (3.9)), i.e., it becomes an NL-point correlation function, where NL is the number of correlated sites.
However, gauge symmetric, conserving approximations, which avoid any spurious condensate amplitudes and, hence, preserve the infrared properties, can still be constructed when the Fock space projection is done in an approximate way. In this approach, first proposed in , the A-integration of (3.8) is done in SPA, while all auxiliary particle Green's functions are derived as pure fluctuation propagators from a generating Luttinger-Ward functional. The A saddle point can be shown to be equivalent to fixing iA as a real chemical potential for the thermodynamic average of the local charge, (Q). Spurious slave boson condensation is avoided by the fact that the fluctuation part of the Bose propagator acquires finite, negative spectral weight Ae>d(!) < 0 for negative frequencies, ! < 0, such that the occupation number density, b(!) Ae>d(!), remains nonnegative for all ! , with b(!) = l/(e^! — 1) the Bose-Einstein distribution. The structure of the self-consistent integral equations for the pseudoparticle propagators is not altered by this approximation. In fact, it may be shown explicitly and in a straightforward way along the lines of [70,71] that the A saddle-point projection preserves the infrared exponents on the level of the simplest conserving approximation, the NCA (to be discussed in the next section). It may be conjectured that this remains true also for more sophisticated conserving approximations. Since the A SPA involves only the thermal average (Q), it is straightforwardly generalized to lattice problems, with a spatially homogeneous chemical potential iA e R. The method has been applied to the Heisenberg lattice in pseudofermion representation in .
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