Photonic solvers capable of finding the ground states of various spin Hamiltonians can be employed to investigate numerous interactive physical systems and combinatorial optimization problems. We introduce a novel photonic solver that leverages spin Hamiltonians within the framework of Modulated Momenta of Light. We establish a real-and-momentum space correspondence for spin Hamiltonians using spatial light transport. The real-space spin interaction is determined by modulating the momentum-space flow of light. This principle, formulated as a generalized Plancherel theorem, enables us to construct a straightforward optical simulator capable of finding ground states for displacement-dependent spin interactions. In our experiments, we apply this principle to unveil the exotic magnetic phase diagram of iron chalcogenides, a family of iron-based superconductors, and we also observe vortex-mediated Berezinskii-Kosterlitz-Thouless dynamics. The precision achieved by momentum modulation in controlling spin interactions holds promise for exploring novel physical effects and solving complex optimization problems.
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