Fig. 7: Out of distribution generalization on the generalized J₁-J₂ Heisenberg model.

a Generalized J₁-J₂ Heisenberg model on a square lattice, with two couplings, J_2L/J₁ and J_2R/J₁. When J_2L = J_2R, the model reduces to the standard J₁-J₂ Heisenberg model. The FNQS is trained on configurations with frustration on only one diagonal and tested on configurations where both diagonals are frustrated. b Spin-spin correlation function on a 6 × 6 lattice of a FNQS trained on \({{{\mathcal{R}}}}=1000\) different realizations of frustration affecting only one of the two diagonals of a square lattice (see Fig. 7 for a schematic representation). The model is tested on two in-distribution points (J_2L/J₁, J_2R/J₁) = (0.0, 0.3) and (0.3, 0.0), and on the out-of-distribution point (J_2L/J₁, J_2R/J₁) = (0.3, 0.3), where both diagonals are frustrated. For reference, the exact results of the J₁-J₂ Heisenberg model at J₂/J₁ = 0.3 are also shown. Inset: The red line shows how the spin-spin correlations are ordered in the panel (b). c Out of distribution generalization of the square magnetization associated with Néel antiferromagnetic order for the FNQS (green circles) as a function of the frustration ratio J₂/J₁ on a 6 × 6 cluster. Exact results (blue line) are shown for reference.