In this work, we proposed and experimentally demonstrated a photonic simulator for $XY$ models, which is a typical kind of classical spin model. By encoding the $XY$ spins on the phase term of the input light field, the corresponding $XY$ Hamiltonian could be performed on the output light intensities. The simulator is mainly based on a programmable and reconfigurable optical vector-matrix multiplication system, which can map arbitrary $XY$ models within the dimensionality limit. Here, we demonstrated the Berezinskii-Kosterlitz-Thouless transition in a two-dimensional $XY$ model, in which the values of some observables are calculated and consistent with the theory. Besides, we performed the ground-state search of two 25-spin $XY$ models with different spin connections and coupling strengths. Our proposal paves the way to investigate the $XY$ spin system.

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• BKT transition
• NP-hard problems
• Photonics
• Classical spin models
• Devices
• Optical computing
• XY model