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In this work, we provide a detailed derivation of Foldy-Wouthuysen (FW)transformation for two-dimensional (2D) gapped Dirac fermions under externalfields and apply the formalism to study valley excitons in 2D semiconductingmaterials. Similar to relativistic quantum few-body problem, the gapped Diracequation can be transformed into a Schr\"{o}dinger equation with "relativistic"correction terms. In this 2D materials system, the correction terms can beinterpreted as the Berry-curvature effect. The Hamiltonian for a valley excitonin external fields can be written based on the FW transformed DiracHamiltonian. Various valley-dependent effects on excitons, such asfine-structure splittings of exciton energy levels, valley-selected excitontransitions, and exciton valley Zeeman effect are discussed within thisframework.

Foldy-Wouthuysen transformation for gapped Dirac fermions in two-dimensional semiconducting materials and valley excitons under external fields