We apply the correlation matrix Hamiltonian reconstruction technique to the two-dimensional Gutzwiller-projected Fermi sea and $\pi$-flux states on finite-sized square and triangular lattices. Our results indicate no spin Hamiltonian with simple local interaction terms stabilizes such states for finite system sizes. We develop a quantitative assessment of the importance of local interactions to the stabilization of these liquid states. Lastly, we systematically assess arguments for the origin of local terms driving a Gutzwiller-projected ground state.

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• Quantum spin liquid
• Strongly correlated systems
• Exact diagonalization
• Lattice gauge theory
• Nonperturbative methods