Based on the finite discretization of the output space, a novel physics-informed neural operator based on the spectral method is developed. Utilizing the Lippmann-Schwinger operator in Fourier space, we construct physical constraints with minimal computational overhead, avoiding the need for automatic differentiation. The latter is achieved by employing a fixed discretization in Fourier space, SPiFOL significantly accelerates training and prediction processes. SPiFOL is trained without labeled data, minimizing equilibrium conditions in Fourier space under macroscopic loading, ensuring periodicity, and efficiently predicting full-field strain responses for arbitrary microstructures. The trained network accelerates FFT solvers by two orders of magnitude while maintaining an average error of less than 1%, even at high phase contrast ratios. We compare several different architectures for SPiFOL, including simple feed-forward neural networks and Fourier neural operator architecture. Additionally, we investigate SPiFOL’s zero-shot super-resolution capabilities for heterogeneous domains and demonstrate its effectiveness on GPUs, achieving substantial reductions in computation time. This work underscores SPiFOL’s potential as a fast, scalable solver for microstructural analysis in material science.