  We propose a general variational fermionic many-body wavefunction that
generates an effective Hamiltonian in a quadratic form, which can then be
exactly solved. The theory can be constructed within the density functional
theory framework, and a self-consistent scheme is proposed for solving the
exact density functional theory. We apply the theory to structurally-disordered
systems, symmetric and asymmetric Hubbard dimers, and the corresponding lattice
models. The single fermion excitation spectra show a persistent gap due to the
fermionic-entanglement-induced pairing condensate. For disordered systems, the
density of states at the edge of the gap diverges in the thermodynamic limit,
suggesting a topologically ordered phase. A sharp resonance is predicted as the
gap is not dependent on the temperature of the system. For the symmetric
Hubbard model, the gap for both half-filling and doped case suggests that the
quantum phase transition between the antiferromagnetic and superconducting
phases is continuous.
