Topological theory of electron-phonon interactions in high-temperature superconductors

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There are large isotope effects in the phonon kinks observed in angle-resolved photoemission spectroscopy of optimally doped cuprate high-temperature superconductors (HTSC), but they are quite different from those expected for a nearly free-electron metal (Fermi liquid) with strong electron-phonon interactions (Eliashberg model). These differences, together with many other anomalies in infrared spectra, seem to suggest that other particles (such as magnons) must be contributing to HTSC. Here we use topological methods to discuss the data, emphasizing nanoscale phase separation and the importance of a narrow band of quantum percolative states near the Fermi energy that is spatially pinned to a self-organized filamentary dopant array, resulting in a filamentary glass. Topological discrete, noncontinuum, nonperturbative methods have previously explained the form of HTSC phase diagrams without involving detailed microscopic assumptions, and they are especially useful in the presence of strong nanoscale glassy disorder. These methods also explain the ``miracle'' of an ideal nearly free-electron (gas or liquid) phonon kink in sharply defined nodal quasiparticle states in LSCO at the metal-insulator transition. Careful study of the data reveals anharmonic phonon interactions. Finally, the universality of the kink energy and Fermi velocities below the Debye cutoff in different cuprates is the result of the marginally elastic nature of these configurationally glassy materials, and specifically the isostatic character of the ${\mathrm{CuO}}_{2}$ planes.