Neural Bloch Eigensolver for Honeycomb Lattices

Haaris Mian

Keywords: neural PDE solvers, eigenvalue problems, physics-informed neural networks, Bloch eigenfunctions, periodic potentials, quantum mechanics

Abstract: This work presents a physics-informed approach to solving the Schrödinger eigenvalue problem associated with particles in two-dimensional periodic potentials, with a focus on honeycomb lattice geometry, due to its non-trivial band topology and relevance to materials such as graphene. By leveraging neural networks to learn complex Bloch functions and their associated eigenvalues (energies) simultaneously, we develop a mesh-free neural solver enforcing the governing PDE, Bloch periodicity, and normalization constraints through a composite loss function without supervision. The model is numerically benchmarked against traditional plane-wave expansion methods. We also explored transfer learning to adapt the solver from nearly-free electron potentials to strongly varying potentials, demonstrating its ability to capture changes in band structure topology. This work contributes to the growing field of physics-informed machine learning for quantum eigenproblems, providing insights into the interplay between symmetry, band structure, and neural architecture.

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Submission Number: 110