In this chapter, we use a notebook to compute the average sign of the square-lattice Hubbard model at as a function of chemical potential, inverse temperature, and lattice size. We will see that simulations can remain feasible in the presence of a sign problem, provided that the average sign is not too small. We will also explore how strongly the sign problem depends on the choice of Hubbard-Stratonovich transformation.
The first section introduces complex configuration weights, reweighting, and the exponential scaling associated with a small average sign. The second section uses pyALF to investigate the average sign numerically.