Graph Neural Networks for Surface-State Prediction in Topological Semimetals: Principles, Architectures, and Emerging Applications

Topological semimetals—encompassing Weyl semimetals, Dirac semimetals, and nodal-line semimetals—host surface states whose existence is guaranteed by bulk topology rather than surface termination details. Predicting these surface states accurately and efficiently is essential for connecting theoretical classifications to experimental observables such as ARPES spectra, quantum oscillations, and anomalous transport coefficients. Classical first-principles approaches are accurate but computationally expensive; they struggle to scan parameter spaces, disorder effects, and heterostructure geometries at scale. Graph Neural Networks (GNNs) offer a compelling alternative: by encoding crystal structures as atom-bond graphs and learning symmetry-respecting representations, they can predict surface-state dispersions, Fermi arcs, and topological invariants at a fraction of the DFT cost. This review systematically examines the theoretical underpinnings of topological surface states, the design principles of GNN architectures suited to this task, benchmark comparisons with DFT, and case studies spanning Weyl semimetals, nodal-line systems, and magnetic topological semimetals relevant to spin transport. We identify open challenges—including disorder, strong correlations, and finite-temperature dynamics—and propose directions for next-generation models.