This study develops a robust analytical and numerical framework to quantify electron-phonon coupling strength \(\lambda\) in topological edge states of two-dimensional (2D) materials under uniaxial strain. We employ a warped Dirac Hamiltonian. It accounts for hexagonal warping, strain-induced lattice distortions, and anharmonic effects. We derive a universal expression for \(\lambda\) . It is valid up to 7% strain with nonlinear corrections. The model predicts 25–70% enhancements in \(\lambda\) compared to bulk-averaged methods, validated by experimental data for Bi \(_2\) Se \(_3\) ( \(\lambda =0.27-0.31\) from 0-5% strain) and twisted bilayer graphene ( \(\lambda \approx 1.2\) ). It integrates electron-electron correlations via dynamical screening, non-perturbative strain effects, and angle-resolved contributions from longitudinal and transverse acoustic phonons, incorporating recent findings on trion binding and chiral phonons. This edge-centric approach elucidates strain-driven quantum phases, such as superconductivity and topological transitions, enabling precise engineering of advanced 2D material technologies like spintronics and quantum devices.
