Complex wave function, Chiral spin order parameter and Phase Problem

We study the two dimensional Hubbard model by use of the ground state algorithm in the Monte Carlo simulation. We employ complex wave functions as trial function in order to have a close look at properties such as chiral spin order ($\chi$SO) and flux phase. For half filling, a particle-hole transformation leads to sum rules with respect to the Green's functions for a certain choice of a set of wave functions. It is then analytically shown that the sum rules lead to the absence of the $\chi$SO. Upon doping, we are confronted with the sign problem, which in our case %ch appears as a ``phase problem" due to the phase of the Monte Carlo weights. The average of the phase shows an exponential decay as a function of inverse temperature similarly to that of sign by Loh Jr. et. al. . We compare the numerical results with those of exact numerical calculations.