  Quantum Monte-Carlo (QMC) simulations involving fermions have the notorious
sign problem. Some well-known exceptions of the auxiliary field QMC algorithm
rely on the factorizibility of the fermion determinant. Recently, a fermionic
QMC algorithm [1] has been found in which the fermion determinant may not
necessarily factorizable, but can instead be expressed as a product of complex
conjugate pairs of eigenvalues, thus eliminating the sign problem for a much
wider class of models. In this paper, we present general conditions for the
applicability of this algorithm and point out that it is deeply related to the
time reversal symmetry of the fermion matrix. We apply this method to various
models of strongly correlated systems at all doping levels and lattice
geometries, and show that many novel phases can be simulated without the sign
problem.
