Non-perturbative methods for metallic quantum criticality and beyond

金属材料中许多引人入胜的现象可能与零温下的“量子相变”(QPT)有关。例如，QPT附近不同微观有序倾向的竞争可能是高温超导体中神秘的消失电阻的原因，但由于强相互作用的影响，这些QPT的理论描述相当具有挑战性，到目前为止主要依赖于微扰方法。也就是说，引入了一个人为的小参数来控制计算。这些方法有自然的局限性，甚至可能导致定性上的错误预测。为了取得进展，我将以两个攻击点为基础，为“非摄动型”工具的开发做出贡献：作为起点，我将研究金属中QPT的一个现实模型，与密度调制的开始有关。我将应用一种新的分析技术，名为“相互作用驱动的尺度”，它将相互作用效应集中在一起；因此，正如我的初步计算所表明的那样，应该会出现一个实际的(非人为的)小参数。这打开了在物理上相关的低能量极限下精确求解该模型的令人兴奋的前景。我将对实验观测进行预测，并将该技术推广到其他分析应用。其次，我将在半数值方法“泛函重整化群”(FRG)的背景下探索“相互作用驱动的标度”，该方法也不需要人为的小参数。此外，我计划应用最近开发的一个名为“多循环FRG”的扩展，它特别适合以一种公正的方式理解相互竞争的排序趋势。因此，通过将它应用于已知存在QPT的电子的简化模型，我希望有助于更好地理解非常规超导电性。

Many fascinating phenomena in metallic materials are possibly connected to a "quantum phase transition" (QPT) at zero temperature. For example, the competition of different microscopical ordering tendencies near a QPT could be responsible for the mysterious vanishing resistance in high-temperature superconductors.The theoretical description of these QPTs is quite challenging due to strong interactions effects, and mostly relied on perturbative methods so far. That is, one introduces an artificial small parameter to render computations controlled. These methods have natural limitations, and might even lead to qualitatively wrong predictions.To make progress, I will contribute to the development of "non-peturbative" tools, with two points of attack: As a starting point, I will study a realistic model of a QPT in a metal, related to the onset of a density modulation. I will apply a novel analytical technique, entitled "interaction-driven scaling", which puts interaction effects in focus; as a result, an actual (non-artificial) small parameter should emerge, as indicated by my preliminary computations. This opens up the exciting prospect of solving the model exactly in the physically relevant limit of low energies. I will make predictions for experimental observables, and will also generalize the technique to other analytical applications. Second, I will explore the "interaction-driven scaling" in the context of the semi-numerical method "functional renormalization group" (fRG), which also does without an artificial small parameter. Furthermore, I plan the application of a recently developed extension called "multiloop fRG" , which is particularly suited to understand competing ordering tendencies in an unbiased fashion. By applying it to a simplified model of electrons known to host a QPT, I therefore hope to contribute to a better understanding of unconventional superconductivity.