Toward the Thermodynamic Limit: Neural Operators for Non-equilibrium Dynamics of Mott Insulators

Abstract

Mott insulators exhibit complex photoexcitation dynamics under intense optical driving, with potential implications for carrier multiplication beyond the Shockley-Queisser limit. Probing these nonequilibrium processes requires access to the thermodynamic limit, where the number of lattice sites becomes arbitrarily large, but conventional solvers are constrained to small systems due to the exponential growth of the Hilbert space. Fourier Neural Operators (FNOs), originally developed for solving partial differential equations, naturally accommodate inputs of varying resolution and are capable of capturing nonlocal effects. Here, we employ FNOs to learn the mapping from noise-perturbed ground-state momentum distributions to their post-pulse counterparts across a range of interaction strengths and driving parameters. Trained only on small lattices, the model generalizes zero-shot to much larger systems, producing physically reasonable momentum distributions well beyond the reach of numerical solvers. Specifically, the model can predict momentum distribution for a 1024x1024 system within a few seconds that matches the theoretical behavior of key observables, whereas direct numerical simulations have so far been restricted to edge sizes of ~30. These results demonstrate the potential of neural operators to directly access large-scale nonequilibrium dynamics, providing a new pathway toward the thermodynamic limit in strongly correlated materials.

Figures (6)

• Figure 1: (a) Runtime of the numerical solver, FNO inference, and ViT inference for one noise realization on an Nvidia RTX 5080 GPU. (b) The predicted average momentum density, $\langle n_{N}\left(\textbf{k}, 200\text{ fs}\right)\rangle$, as a function of the system size, $N$. The FNO correctly stabilizes toward a limiting value as $N\to\infty$, while the ViT does not. Here, $U/\tau=3.497$, and a pulse of $A=0.5$ MV/cm and $\omega=10$ THz is used. An approximate initial state is used due to the computational limitations of the numerical simulation. (c) The difference between the predicted average momentum density from (b) and its approximate limiting value, $c$, as $N$ becomes large. For the ViT, this limiting value is the average momentum density at $N=148$. For the FNO, this limiting value is the average momentum density at $N=2048$. The FNO converges stably toward a limiting value with a power law scaling.
• Figure 2: Example predictions of the post-pulse momentum distribution for various system sizes. All but the leftmost column are predictions on systems larger than those in the training data. All but the rightmost column are initial states that are generated using our numeric simulation. The last column is an approximation of an initial state in which the Brillouin zone is filled with a small constant value everywhere except for a target momentum excitation. The first row is the initial state at $t=0$ fs, the second row is the FNO prediction at $t=200$ fs, and the third row is the ViT prediction at $t=200$ fs. No ground truth is included, as the computation cost is intractable to run real-time evolution for the two larger systems. The ViT exhibits unphysical artifacts, circled in green.
• Figure 3: Randomly-selected post-pulse momentum distribution predictions on system sizes where the ground truth is accessible. Both models accurately predict the ground truth within the range of system and pulse parameters that were trained on.
• Figure 4: Relative $L_2$ test loss at various system sizes when the model is provided $1/\sqrt{N}$, $1/N$, or $1/N^2$ as input. Training was performed for 100 epochs with a scheduler patience of 5, and all other hyperparameters were the same.
• Figure 5: Example predictions on various system sizes when the model is provided $1/\sqrt{N}$, $1/N$, and $1/N^2$ as input. The predictions are very similar.