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Enhanced electron-phonon coupling near an electronic quantum phase transition

I construct a simple model to demonstrate that when the many-electron quantum state of a material is near a quantum phase transition and the vibrational motion of a phonon explores the potential energy surface near the transition point, then an impenetrable barrier appears at the potential energy surface which restricts the phonon from crossing the transition point and abnormally increases the phonon frequency. The ensuing anomalous enhancement of the electron phonon coupling is general and independent of the specific nature of the electronic quantum phase transition. Understanding and modelling this strong electron-phonon coupling may potentially lead to the design of phonon superconductors with high critical temperatures by choosing their parameters appropriately near an electronic quantum phase transition.