  Optical lattice experiments with ultracold fermion atoms and quantum gas
microscopy have recently realized direct measurements of magnetic correlations
at the site-resolved level. We calculate the short-range spin correlation
functions in the ground state of the two-dimensional repulsive Hubbard model
with the auxiliary-field Quantum Monte Carlo (AFQMC) method. The results are
numerically exact at half filling where the fermion sign problem is absent.
Away from half-filling, we employ the constrained path AFQMC approach to
eliminate the exponential computational scaling from the sign problem. The
constraint employs unrestricted Hartree-Fock trial wave-functions with an
effective interaction strength U , which is optimized self-consistently within
AFQMC. Large supercells are studied, with twist averaged boundary conditions as
needed, to reach the thermodynamic limit. We find that the nearest-neighbor
spin correlation always increases with the interaction strength U , contrary to
the finite-temperature behavior where a maximum is reached at a finite U value.
We also observe a change of sign in the next nearest neighbor spin correlation
with increasing density, which is a consequence of the buildup of the
long-range anti-ferromagnetic correlation. We expect the results presented in
this work to serve as a benchmark as lower temperatures are reached in
ultracold atom experiments.
