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性质的许多了解都是基于单粒子图像。也就是说，一旦我们确定了一个电子的作用，那么任何宏观电流或响应都是由这种单粒子行为决定的，只需稍作修改就可以解释许多电子的统计数据。虽然这个简单的想法让我们能够理解简单金属的性质，但它提供了对具有相关性质的金属以及具有相关电子行为的可能的态的可能跃迁的有限理解。一个多世纪前发现的一种这样的状态就是超导。它具有惊人的性质，尤其是无限的导电性和完美的抗磁性。这种非凡的物质状态的理论到目前为止已经被“理解”了50多年。我在引语中使用“理解的”是为了强调它是在一定程度上被理解的，但我们需要超越这个水平，以实现一个真正的预测理论，一个将把超导与其他不稳定性区分开来的理论。这就是本研究提案中描述的工作将完成的工作。过去的工作集中在超导本身的机制上，即电子-声子相互作用。它在很大程度上忽略了解释中的竞争者状态，也在很大程度上忽略(或至少过于简化)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，即该理论将告诉我们，太强的耦合强度将导致超导性以外的状态。这种自我调节的Eliashberg理论不仅将预测其他状态，还将标志着电子-声子相互作用的局限性。这些限制来自各种可能的来源，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符合准确结果的框架(例如量子蒙特卡罗)的输入，以便能够对其他框架(例如Eliashberg理论)所需的近似进行控制。一旦达到这一控制水平，我们预计理论与实验的合作将比目前的密度泛函理论(DFT)更透明。事实上，我们希望通过这项研究中获得的见解来改进DFT配方。通过这种方式，我们有望对超导电性及其竞争态有一个更统一和完整的描述，从简单的金属到包括铜酸盐和钠化合物在内的不同家族，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.