Functional Integrals for Correlated Electrons

The application of functional integral methods and the Hubbard--Stratonovich transformation to the Hubbard model is discussed. For the attractive case, using a simple gauge transformation of the superconducting order parameter field, the effective action for the low-energy phase excitations is derived. Inclusion of the electromagnetic field is straightforward. For the repulsive case at half--filling use of a fluctuating spin reference frame allows a derivation of the effective nonlinear sigma model for arbitrary coupling strength U. Results for the spin-wave velocity and the spin stiffness are obtained for arbitrary U. Away from half--filling, at large U an effective action for doped fermions and the spins is found which is related to the slave fermion formulation of the correlated fermion problem. The formalism is generalized to include the full SU(2)xSU(2) symmetry of the Hubbard model. This allows the derivation of a Ginzburg-Landau like theory that includes both particle-hole and particle-particle excitations. (paper at the NATO workshop on the Hubbard model, San Sebastian, Spain, oct. 1993)