Interwoven electronic and phononic topologies via electron–phonon coupling and locality-engineered edge states in monolayer ZrBr

Monolayer ZrBr combines excellent electronic and phononic transport, forming a compact platform for coupled topology. First-principles and Wannier-Green's-function calculations show symmetry-protected near-linear crossings along K-Γ-M-K without spin-orbit coupling (SOC). With SOC, a direct gap of ∼45.6 meV opens at Γ with a pronounced reorganization of Zr-4d orbital weights. The Wilson-loop evolution yields Z₂ = 1, and semi-infinite boundary spectra exhibit a gap-traversing helical edge channel, consistent with the constant-energy contour and in-plane spin winding. Phonon dispersions contain no imaginary frequencies and resolve two low-frequency softening anomalies at Γ and near (0.09,0.16,0); their progressive suppression with increasing electronic smearing indicates a Kohn-anomaly-type renormalization driven by Fermi-surface screening. Atom-projected phonon densities of states partition vibrations into a Zr-dominated low-frequency sector and a Br-dominated mid-to-high-frequency sector. The Eliashberg function α²F(ω) yields strong coupling λ = 1.26 and superconducting quasiparticle signatures with T_c = 21.67 K. Around ∼3.45 THz, boundary phonon spectra, iso-frequency contours, and sign-alternating phonon Berry curvature jointly indicate a nontrivial phonon geometric response.