Reverse Engineering Hamiltonians from Wavefunctions

Effective models in quantum materials are typically constructed by truncating the full Hilbert space to a few Wannier orbitals, but such truncations often distort the quantum geometry of the low-energy band. We introduce a framework to reverse engineer effective Hamiltonians directly from wavefunctions by optimally preserving the form factors. Our approach is based on the Mercer decomposition of the overlap kernel that provides the optimal low-rank approximation to the Bloch band of interest. The resulting truncated states define effective Hamiltonians that capture both spectral and geometric features of the full system.