Instanton gas approach to the Hubbard model

In this paper, we consider a path integral formulation of the Hubbard model based on a Hubbard-Stratonovich transformation that couples the auxiliary field to the local electronic density. This decoupling is known to have a saddle-point structure that shows a remarkable regularity: The field configuration at each saddle point can be understood in terms of a set of elementary field configurations localized in space and imaginary time which we coin instantons. The interaction between instantons is short ranged. Here, we formulate a classical partition function for the instanton gas that has predictive power. For a given set of physical parameters, we can predict the distribution of instantons and show that the instanton number is sharply defined in the thermodynamic limit, thereby defining a unique dominant saddle point. Decoupling in the charge channel conserves SU(2) spin symme-try 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.