Beyond the instanton gas approach: dominant thimbles approximation for the Hubbard model

To each complex saddle point of an action, one can attach a Lefschetz thimble on which the imaginary part of the action is constant. Cauchy theorem states that summation over a set of thimbles produces the exact result. This reorganization of the path integral, is an appealing starting point for various approximations: In the realm of auxiliary quantum Monte Carlo methods it provides a framework to alleviate the negative sign problem. Here, we suggest to constrain the integration to the \textit{dominant} thimbles: the thimbles attached to the saddle points with the largest statistical weight. For the Hubbard model, in a formulation where the the Hubbard Stratonovitch field couples to the charge, this provides a \textit{symmetry} consistent approximation to the physics of the Hubbard model: constraining the integration domain does not explicitly break a symmetry. We can test this approach for the Hubbard model at half-filling on a bipartite lattice. The paper builds on the previously developed instanton gas approach, where an exhaustive saddle point approximation was constructed. We present results, showing that the dominant thimbles approximation provides results that are in remarkable agreement with the exact results for various fermionic observables including spin and charge order parameters and single electron spectral functions. We discuss implications of our results for simulations away of half filling.