Leveraging recurrence in neural network wave functions for large-scale simulations of Heisenberg antiferromagnets

Machine-learning-based variational Monte Carlo simulations are a promising approach for targeting quantum many body ground states, especially in two-dimensions and in cases where the ground state is known to have a non-trivial sign structure. While many state-of-the-art variational energies have been reached with these methods for finite-size systems, little work has been done to use these results to extract information about the target systems in the thermodynamic limit. Here, we employ recurrent neural networks (RNNs) as our variational ansatzë, because the recurrent nature of the architecture allows one to iteratively retrain the same network for progressively larger physical systems. This transfer learning technique, which has been termed ``iterative retraining'', is unique to RNN wave functions and allows us to study lattices with L up to 36 without beginning optimization from scratch for each system size. More specifically, we study Heisenberg antiferromagnets on square and triangular lattices with open and periodic boundary conditions and we perform finite-size scaling studies of our variational energies and of the spin structure factor. For the square lattice, we carefully benchmark our results and show that they are in good agreement with existing literature. For the triangular lattice, for which there are fewer benchmarks available, we find a non-vanishing extrapolation of the structure factor, which is indicative of the expected long-range order of the ground state. These results demonstrate how the unique ability of RNN wave functions to generalize in physical system size allows us to study the physics of the target system in the thermodynamic limit.