Superconductivity and competing states: the role of the electron-phonon and Coulomb interactions

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Much of what we understand about the electronic properties of simple metals and semiconductors is based on a single-particle picture. That is, once we determine what one electron does, then any macroscopic current or response is determined by this single-particle behaviour, with slight modifications to account for the statistics of the many electrons. While this simple idea allows us to understand the properties of simple metals, it provides limited understanding for metals with correlated properties, and for the possible transitions to states with correlated electron behaviour. One such state, discovered more than a century ago, is superconductivity. It has dramatic properties, not least of which are infinite conductivity and perfect diamagnetism. The theory for this remarkable state of matter has been "understood" for more than 50 years now. I use "understood" in quotations to emphasize that it is understood at a certain level, but we need to go beyond this level, to achieve a truly predictive theory, and one which will delineate superconductivity from other instabilities. That is what the work described in this research proposal will accomplish. Past work has focussed on the mechanism for superconductivity itself, the electron-phonon interaction. It largely ignored the competitor states in the explanation, and it largely ignored (or at least oversimplified) the role of direct Coulomb interactions in the explanation as well. We will formulate a theory that is self-regulating, i.e. the theory will tell us that too strong a coupling strength will result in states other than superconductivity. This self-regulating Eliashberg theory will not only predict other states, but will signal the limitations of the electron-phonon interaction. These limitations come from a variety of possible sources, including both single-electron effects (the electron acquires a very large effective mass through the electron-phonon interaction), and through direct repulsive interactions between two electrons. This work will require input from frameworks more amenable to exact results (e.g. Quantum Monte Carlo) so that control over the approximations required in other frameworks (e.g. Eliashberg theory) can be exercised. And once this level of control is achieved, we expect the theory to work with experiment in a much more transparent way than is currently done i.e. with Density Functional Theory (DFT). In fact, we hope to improve DFT formulations through insights gained in this research. In this way we will hopefully have a more unified and complete description of superconductivity and its competitor states, amongst different families ranging from the simple metals, and including the cuprates and the pnictides, right through to the high-pressure hydrides.