Mode-Resolved Electron-Phonon Topology from Lithium to Cesium

We present a unified, mode-resolved framework for electron-phonon entanglement and symmetry protection in elemental alkali metals, using Li as the working example and  Na, K, Rb, and Cs for comparison. The mode-resolved band-curvature (“second-order”) electron-phonon band structures that track how individual normal modes modulate adiabatic band energies, we build a k-space map that pinpoints where normal modes can or cannot lift electronic degeneracies. In BCC Li, symmetry-protected crossings exhibit equal-and-opposite spikes, indicating purely off-diagonal, nonadiabatic mixing that forbids gap opening and helps suppress superconductivity. Lower-symmetry polymorphs (9R, cl16) display a broad constant curvature region following a parabolic dependence on the normal-mode coordinate. Under pressure, FCC Li concentrates strong coupling within a narrowly confined pocket of reciprocal space (near X). Generalizing across the alkali series in the BCC phase, we find a universal feature: pole-like, antisymmetric curvature localized to the H→N line of the BCC Brillouin zone, consistent with interband mixing within quasi-degenerate doublets. A minimal Dirac (two-level) model reproduces these phonon-tunable gaps while preserving the underlying linear dispersion. Quantizing the mode coordinate yields a spin–boson description: for Li, K, Rb, and Cs, the maximum mode-induced splitting (degeneracy-lifting energy) is comparable to the longitudinal phonon energy, indicating near-resonant coupling, which explains the large, equal-and-opposite curvature poles and coherent interband mixing we observed. In contrast, Na is off-resonant, with its degeneracy-lifting energy 4.5 times smaller than its longitudinal phonon energy. In summary, these results motivate symmetry-targeted control of lattice spectra and near-degenerate band splittings (via pressure, alloying, temperature) to tune effective masses, coherence, and pairing tendencies with implications for superconductivity, analog quantum simulation, and lattice-embedded quantum sensors.