  We present a guiding principle for designing fermionic Hamiltonians and
quantum Monte Carlo (QMC) methods that are free from the infamous sign problem
by exploiting the Lie groups and Lie algebras that appear naturally in the
Monte Carlo weight of fermionic QMC simulations. Specifically, rigorous
mathematical constraints on the determinants involving matrices that lie in the
split orthogonal group provide a guideline for sign-free simulations of
fermionic models on bipartite lattices. This guiding principle not only unifies
the recent solutions of the sign problem based on the continuous-time quantum
Monte Carlo methods and the Majorana representation, but also suggests new
efficient algorithms to simulate physical systems that were previously
prohibitive because of the sign problem.
