Instanton gas approach to the Hubbard model

Decoupling in the charge channel conserves SU(2) spin symmetry for each field configurations. Hence, the instanton approach provides an SU(2) spin-symmetric approximation to the Hubbard model. It fails, however, to capture the magnetic transition inherent to the Hubbard model on the honeycomb lattice despite being able to describe local moment formation.

In fact, the instanton itself corresponds to local moment formation and concomitant short-ranged antiferromagnetic 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 thimble approaches to alleviate the sign problem are also discussed.