Transformer-Based Neural-Network Quantum State Method for Electronic Band Structures of Real Solids.

Recent advancements in neural networks have led to significant progress in addressing many-body electron correlations in small molecules and various physical models. In this work, we propose QiankunNet-Solid, which incorporates periodic boundary conditions into the neural network quantum state (NNQS) framework based on generative Transformer architecture along with a batched autoregressive sampling (BAS) method, enabling the effective ab initio calculation of real solid materials. The accuracy of this method is demonstrated in one-, two-, and three-dimensional periodic systems, with results comparable to those of full configuration interaction and coupled-cluster method, even in the strongly correlated regime. Furthermore, we compute the band structures and density of states for silicon crystal. The successful incorporation of periodic boundary conditions into the NNQS framework through QiankunNet-Solid opens up new possibilities for the accurate and efficient study of electronic structure properties in solid-state physics.