Fluctuating field series: Towards calculations of correlated systems with high accuracy

We introduce regular series expansion for weakly and moderately correlated fermionic systems based on the fluctuating local field approach. The method relies on explicitly accounting for the leading fluctuating mode(s) and is therefore suitable for medium-sized lattices. It can also be used as a solver for the cluster approximations for infinite-size lattices. We introduce a classical fluctuating field coupled to a fermionic collective mode(s). In this way, fluctuations in selected modes are treated in a nonperturbative way. Other degrees of freedom are accounted for in the diagram expansion performed for each value of the fluctuating field. The method is benchmarked for the $U/t=1$ and $U/t=2$ Hubbard lattices at half filling. Results for susceptibility in the antiferromagnetic channel along with the spectral function are compared with the numerically exact quantum Monte Carlo data. Calculations up to the third order of the series expansion are performed and show a uniform convergence to the reference result for the susceptibility. This convergence is observed well below the dynamical mean-field theory (DMFT) Ne\'el temperature and makes our practically simple method applicable in a much wider temperature range than DMFT-based diagrammatic schemes.