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

我们对简单金属和半导体的电子特性的了解大多是基于单粒子的图像。也就是说，一旦我们确定了一个电子的行为，那么任何宏观电流或反应都是由这个单粒子的行为决定的，并对许多电子的统计数据进行了轻微的修改。虽然这个简单的想法使我们能够理解简单金属的性质，但它对具有相关性质的金属以及具有相关电子行为的状态的可能转变提供了有限的理解。一个多世纪以前发现的一种这样的状态就是超导。它具有惊人的性能，其中最重要的是无限导电性和完美的抗磁性。到目前为止，人们已经“理解”了这种非凡的物质状态的理论50多年了。我在引用中使用“理解”来强调它在一定程度上是被理解的，但我们需要超越这个水平，实现一个真正的预测理论，一个将从其他不稳定性中描绘超导性的理论。这就是本研究计划中所描述的工作将要完成的。过去的工作集中在超导本身的机制，电子-声子相互作用。它在很大程度上忽略了解释中的竞争态，并且在很大程度上忽略了（或者至少是过度简化了）直接库仑相互作用在解释中的作用。我们将制定一个自我调节的理论，即理论将告诉我们，太强的耦合强度将导致超导以外的状态。这种自我调节的Eliashberg理论不仅可以预测其他状态，还可以表明电子-声子相互作用的局限性。这些限制来自多种可能的来源，包括单电子效应（电子通过电子-声子相互作用获得非常大的有效质量），以及两个电子之间的直接排斥相互作用。这项工作将需要来自更适合精确结果的框架（例如量子蒙特卡罗）的输入，以便可以对其他框架（例如Eliashberg理论）所需的近似值进行控制。一旦达到这种控制水平，我们期望该理论以比目前更透明的方式与实验一起工作，即密度泛函理论（DFT）。事实上，我们希望通过本研究中获得的见解来改进DFT公式。通过这种方式，我们有望对超导及其竞争状态有一个更统一、更完整的描述，在不同的金属族中，从简单金属，包括铜酸盐和羟基，一直到高压氢化物。

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.