Lattice relaxation in twisted van der Waals materials is not merely a numerical nuisance; it is the mechanism that sets the electronic landscape. Here, we uncover how relaxation is generated and why it reshapes bands and topology. Starting from continuum elastic theory, we derive closed-form relaxation fields by minimizing intralayer elasticity energy and interlayer adhesion energy. We further introduce an analytical phase factor expansion theory that maps relaxation into the electronic Hamiltonian. The analysis identifies relaxation-induced gaugelike fields that control bandwidth suppression, band inversions, and valley Chern numbers. While our framework yields orders-of-magnitude speedup over supercell-based ab initio workflows, its central contribution is mechanistic: a unified, analytic description that predicts and interprets relaxation-driven band engineering in twisted systems such as twisted graphene and ${\mathrm{MoTe}}_2$. Furthermore, our framework interfaces naturally with many-body methods.

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• Electronic structure
• Twistronics
• 2-dimensional systems
• Twisted heterostructures
• Band structure methods
• Bloch wave theory
• Tight-binding model
• k dot p method