Physics-Informed Neural Solvers for Periodic Quantum Eigenproblems

TL;DR

The work tackles the Floquet-Bloch eigenproblem for particles in two-dimensional periodic potentials, focusing on honeycomb lattices with Dirac-point topology. It adopts a mesh-free physics-informed neural network to learn Bloch functions $u_{n,\mathbf{k}}(\mathbf{x})$ and energies $E_{n,\mathbf{k}}$ by enforcing the Schrödinger equation with the Bloch-modified operator $\mathcal{H}_b = -\tfrac{1}{2}(\nabla + i\mathbf{k})^2 + V(\mathbf{x})$, Bloch periodicity, and normalization in an unsupervised setting across the Brillouin zone. The study introduces a neural-parameterization of Bloch states, a composite loss with PDE residual, normalization, and boundary-condition terms, and validates results against plane-wave expansions, including transfer learning to extend from nearly-free to strongly varying potentials; it also visualizes Bloch modes and analyzes convergence. This framework provides a scalable, mesh-free tool for exploring quantum materials' band topology, enabling rapid adaptation to potential landscapes while preserving essential physical constraints. Overall, the work advances physics-informed learning for multidimensional periodic quantum eigenproblems and offers insights into how symmetry and neural architectures interact to capture Dirac physics in honeycomb lattices.

Abstract

This thesis presents a physics-informed machine learning framework for solving the Floquet-Bloch eigenvalue problem associated with particles in two-dimensional periodic potentials, with a focus on honeycomb lattice geometry, due to its distinctive band topology featuring Dirac points and its 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 solver enforcing the governing Schrödinger equation, Bloch periodicity, and normalization constraints through a composite loss function without supervision. The model is trained over the Brillouin zone to recover band structures and Bloch modes, with numerical validation against traditional plane-wave expansion methods. We further explore transfer learning techniques 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 architectures.

Figures (8)

• Figure 1: Brillouin zone of the honeycomb lattice and reciprocal lattice vectors {$b_1$, $b_2$}. High symmetry points $\Gamma$, K, M, K' labeled, along with the path connecting high-symmetry points over the irreducible wedge of the Brillouin zone.
• Figure 2: A simple honeycomb lattice potential formed by the superposition of three cosine waves, $V(\mathbf{x}) = V_0 \sum_{i=1}^{3} \cos(\mathbf{b_i}\cdot\mathbf{x})$, where $\mathbf{b_i}$ are the reciprocal lattice vectors.
• Figure 3: A side by side comparison of the band structure due to a strong honeycomb potential (left) and the free electron structure (right). The stronger potential induces significant band gaps and clearly show to the characteristic Dirac point at the K points, while the free electron model exhibits parabolic bands without gaps. Dispersion obtained via plane-wave expansion method and analytic solution respectively.
• Figure 4: Sampling strategy in reciprocal space. Points are sampled uniformly within the Irreducible Brillouin Zone (IBZ) and its lattice translated copies to expose the network to higher quasi-momenta and energy bands.
• Figure 5: Band structure learned by the physics-informed neural network plotted along the high-symmetry path $\Gamma \rightarrow K \rightarrow M \rightarrow \Gamma$ in the Brillouin zone for a weak honeycomb potential with $V_0 = 1$. The learned dispersion curves match closely with those obtained from the plane-wave expansion method, and closely approximate the free particle dispersion due to the weak potential.