  Sign problem in fermion quantum Monte Carlo (QMC) simulation appears to be an
extremely hard problem. Traditional lore passing around for years tells people
that when there is a sign problem, the average sign in QMC simulation
approaches zero exponentially fast with the space-time volume of the
configurational space. We, however, analytically show this is not always the
case and manage to find physical bounds for the average sign. Our understanding
is based on a direct connection between the sign bounds and a well-defined
partition function of reference system and could distinguish when the bounds
have the usual exponential scaling, and when they are bestowed on an algebraic
scaling at low temperature limit. We analytically explain such algebraic sign
problems found in flat band moir\'e lattice models at low temperature limit. At
finite temperature, a domain size argument based on sign bounds also explains
the connection between sign behavior and finite temperature phase transition.
Sign bounds, as a well-defined observable, may have ability to ease or even
make use of the sign problem.

Strongly Correlated Electrons;Statistical Mechanics;High Energy Physics - Lattice;High Energy Physics - Theory