A benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo

The ground state properties of the two-dimensional Hubbard model are calculated with the auxiliary-field quantum Monte Carlo (AFQMC) method. With general twist boundary conditions, the shell effect is eliminated. We use large lattice sizes ($L\times L$ lattices with $L$ up to $24$) and average over many twist angles to extrapolate to the thermodynamic limit and ensure convergence of the calculated physical quantities. At half filling, we obtain accurate results for the ground-state energy, sublattice magnetization and double occupancy, and other correlation functions for interactions ranging from $U/t = 2$ to $8$. We then study the doped cases with a constraint to control the ``minus sign'' problem. We apply the latest development \footnote{Hao~Shi, Shiwei~Zhang, Phys. Rev. B. {\bf 88}, 125132 (2013); Hao~Shi, Carlos~A.~Jim\'enez-Hoyos, R.~Rodr\'iguez-Guzm\'an, Gustavo~E.~Scuseria, and Shiwei~Zhang, Phys. Rev. B. {\bf 89}, 125129 (2014) } in the constrained path AFQMC technique to benchmark the effect of different trial wave functions, and study ground-state properties in the thermodynamic limit. The different competing phases close to half filling are investigated.