Non-perturbative methods for metallic quantum criticality and beyond

金属材料中许多有趣的现象都可能与零温度下的“量子相变”有关。例如，QPT附近不同微观有序趋势的竞争可能是高温超导体中神秘消失的电阻的原因。由于这些qpt具有很强的相互作用，理论描述相当具有挑战性，迄今为止主要依赖于微扰方法。也就是说，引入一个人为的小参数来控制计算。这些方法有天然的局限性，甚至可能导致定性错误的预测。为了取得进展，我将为“非扰动”工具的开发做出贡献，有两个切入点：作为起点，我将研究金属中与密度调制开始相关的QPT的现实模型。我将运用一种新的分析技术，名为“互动驱动缩放”，它将互动效应放在焦点上；因此，正如我的初步计算所表明的那样，应该出现一个实际的（非人为的）小参数。这开辟了一个令人兴奋的前景，即在低能量的物理相关极限下精确地解决这个模型。我将对实验观察结果进行预测，并将该技术推广到其他分析应用中。其次，我将在半数值方法“功能重整化群”（fRG）的背景下探索“相互作用驱动的缩放”，该方法也不需要人工的小参数。此外，我计划应用最近开发的名为“multiloop fRG”的扩展，它特别适合以无偏的方式理解竞争排序趋势。因此，通过将其应用于已知承载QPT的简化电子模型，我希望有助于更好地理解非常规超导性。

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.