Flat band projections: Sign problem mapping for frustrated spin systems

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.