This paper presents a new way to design quantum materials by searching directly for the microscopic interactions that produce a desired ground state. The authors build on a recent idea called Foundation Neural-Network Quantum States (FNQS). In this approach a single neural-network wave function depends not just on the particle arrangement but also on the interaction strengths, or couplings, of the model. That lets the researchers treat any property of the ground state as a smooth function of the couplings and then move the couplings downhill by gradient methods to find models with the target property.

A Hamiltonian is the mathematical description of a quantum system; its couplings are the numbers that set the interaction strengths. Here the FNQS ansatz is trained to represent ground states across a family of Hamiltonians at once. At each step the network is trained locally to give good ground states near the current couplings. Then the gradient of the chosen property with respect to the couplings is computed and used to update the couplings. The authors note this gradient can be obtained at the same cost as the usual variational forces used to find a single ground state, so the search avoids a costly brute‑force scan over many candidate Hamiltonians.

The team used this framework to look for nonmagnetic phases of frustrated spin-1/2 Heisenberg models on the square lattice. “Frustrated” here means competing interactions that make it hard for spins to settle into a simple ordered pattern. They first tested the method with one free next‑nearest‑neighbor coupling and recovered a previously known nonmagnetic window of the square J1–J2 Heisenberg model. Letting the two diagonal couplings vary independently revealed a larger nonmagnetic region that links the square‑lattice regime and the anisotropic triangular‑lattice regime. Finally they searched an eight‑dimensional space of couplings inside a 2×2 unit cell. Starting from uniform couplings, the optimization split them into two alternating groups and converged to an effective J1–J2–δ model, where diagonal couplings alternate between two values on neighboring plaquettes.