Effect of the exchange hole on the Gutzwiller approximation in one dimension

The Gutzwiller approximate solution to the Gutzwiller wavefunction yields exact results for the Gutzwiller wavefunction in the infinite dimensional limit. Implicit in the Gutzwiller approximation is an approximate local form of the fermion exchange hole. This approximate form is the same for all dimensions but is incorrect except in infinite dimensions. We implement the correct form for the exchange hole into the Gutzwiller approximation. We perform calculations on the one-dimensional Hubbard model at half-filling. They indicate that the implementation of the exchange hole already brings the Gutzwiller approximation into very close quantitative agreement with the results of the full Gutzwiller wavefunction. Metallicity as well as anti-ferromagnetism are recovered.