  In this article we consider a path integral formulation of the Hubbard model
based on a SU(2)-symmetrical Hubbard-Stratonovich transformation that couples
auxiliary field to the local electronic density. This decoupling is known to
have a regular saddle-point structure: each saddle point is a set of elementary
field configurations localized in space and imaginary time which we coin
instantons. We formulate a classical partition function for the instanton gas
that has predictive power. Namely, we can predict the distribution of
instantons and show that the instanton number is sharply defined in the
thermodynamic limit, thus defining a unique dominant saddle point. Despite the
fact that the instanton approach does not capture the magnetic transition
inherent to the Hubbard model on the honeycomb lattice, we were able to
describe the local moment formation accompanied by short-ranged
anti-ferromagnetic correlations. This aspect is also seen in the single
particle spectral function that shows clear signs of the upper and lower
Hubbard bands. Our instanton approach bears remarkable similarities to local
dynamical approaches, such as dynamical mean field theory, in the sense that it
has the unique property of allowing for local moment formation without breaking
the SU(2) spin symmetry. In contrast to local approaches, it captures
short-ranged magnetic fluctuations. Furthermore, it also offers possibilities
for systematic improvements by taking into account fluctuations around the
dominant saddle point. Finally, we show that the saddle point structure depends
upon the choice of lattice geometry. For the square lattice at half-filling,
the saddle point structure reflects the itinerant to localized nature of the
magnetism as a function of the coupling strength. The implications of our
results for Lefschetz thimbles approaches to alleviate the sign problem are
also discussed.
