Lefschetz thimbles decomposition for the Hubbard model on the hexagonal lattice.

In this article we study the properties of the Lefschetz thimbles decomposition for the Hubbard model on the hexagonal lattice both at zero chemical potential and away from half-filling. We start from the detection of saddle points, for real and complex-valued fields. A Schur complement solver and the exact computation of the derivatives of the fermion determinant are employed. These technical improvements allows us to look at lattice volumes as large as $12 \times 12$ with $N_\tau=256$ steps in Euclidean time, in order to capture the properties of the thimbles decomposition as the thermodynamic, low temperature and continuum limits are approached. Different versions of the Hubbard-Stratonovich (HS) decomposition were studied and we show that the complexity of the thimbles decomposition is very dependent on its specific form. In particular, we demonstrate the existence of an optimal regime, with a reduced number of thimbles becoming important in the overall sum. In order to check these findings, we have performed quantum Monte Carlo (QMC) simulations using the holomorphic gradient flow to deform the integration contour into the complex plane. Several benchmark calculations were made on small volumes ($N_s=8$ sites in space), albeit still at low temperatures and with the chemical potential tuned to the van Hove singularity, thus entering into a regime where standard QMC techniques exhibit exponential decay of the average sign. The results are compared versus exact diagonalization (ED), and we demonstrate the importance of choosing an optimal form for the HS transformation to avoid issues associated with ergodicity. We compare the residual sign problem with the state-of-the-art BSS(Blankenbecler, Scalapino and Sugar)-QMC and show that the average sign can be kept substantially higher using the Lefschetz thimbles approach.