  Photonic solvers that are able to find the ground states of different spin
Hamiltonians can be used to study many interactive physical systems and
combinatorial optimization problems. Here, we establish a real-and-momentum
space correspondence of spin Hamiltonians by spatial light transport. The
real-space spin interaction is determined by modulating the momentum-space flow
of light. This principle is formulated as a generalized Plancherel theorem,
allowing us to implement a simple optical simulator that can find the ground
states for any displacement-dependent spin interactions. Particularly, we use
this principle to reveal the exotic magnetic phase diagram from a J1-J2-J3
model, and we also observe the vortex-mediated Berezinskii-Kosterlitz-Thouless
dynamics from the XY model. These experiments exhibit high calculation
precision by subtly controlling spin interactions from the momentum space of
light, offering a promising scheme to explore novel physical effects.
