  Projection of the Coulomb potential onto flat bands paves the way to design
various interactions in the particle-hole and particle-particle channels. Here
we pose the question if we can use this mapping to overcome the negative sign
problem for the simplest possible frustrated spin system consisting a trimer of
spins-1/2 coupled with an antiferromagnetic exchange interaction. While the
answer is negative, we show that we can map the sign problem for frustrated
spin systems onto a problem where we need to simulate particle-hole symmetric
systems but with long-ranged Coulomb interactions prevailing over short-ranged
ones. While the latter systems are currently not accessible to auxiliary field
determinant quantum Monte Carlo, this mapping motivates algorithmic development
that may overcome this issue.
