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The concept of Lefschetz thimble decomposition is one of the most promisingpossible modifications of Quantum Monte Carlo (QMC) algorithms aimed atalleviating the sign problem which appears in many interesting physicalsituations, e.g. in the Hubbard model away from half filling. In this approachone utilizes the fact that the integral over real variables with an integrandcontaining a complex fluctuating phase is equivalent to the sum of integralsover special manifolds in complex space ("Lefschetz thimbles"), each of themhaving a fixed complex phase factor. Thus, the sign problem can be reduced ifthe resulting sum contains terms with only a few different phases. We explorethe complexity of the sign problem for the few-site Hubbard model on squarelattice combining a semi-analytical study of saddle points and thimbles insmall lattices with several steps in Euclidean time with results of test QMCcalculations. We check different variants of conventional Hubbard-Stratonovichtransformation based on Gaussian integrals. On the basis of our analysis wereveal a regime with minimal number of relevant thimbles in the vicinity ofhalf filling. In this regime we found only two relevant thimbles for thefew-site lattices studied in the paper. There is also indirect evidence of theexistence of this regime in more realistic systems with large number ofEuclidean time slices. In addition, we derive a new non-Gaussian representationof the interaction term, where the number of relevant Lefschetz is also reducedin comparison with conventional Gaussian Hubbard-Stratonovich transformation.

Path integral representation for the Hubbard model with reduced number of Lefschetz thimbles