Non-perturbative methods for metallic quantum criticality and beyond

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.

金属材料中许多引人入胜的现象可能在零温度下连接到“量子相变”（QPT）。例如，QPT附近不同显微镜排序趋势的竞争可能导致高温超导体中神秘消失的抵抗力。这些QPT的理论描述由于强烈的相互作用效果而挑战，并且主要依赖于迄今为止的扰动方法。也就是说，一个人引入了一个人工小参数来渲染受控制的计算。这些方法具有自然的局限性，甚至可能导致定性上错误的预测。要取得进步，我将为“非扰动”工具的开发做出贡献，并具有两个攻击点：作为起点，我将研究金属中QPT的现实模型，与密度调节的开始有关。我将采用一种新颖的分析技术，标题为“相互作用驱动的缩放”，这将相互作用的效果置于焦点；结果，如我的初步计算所示，应出现实际（非人造的）小参数。这打开了令人兴奋的前景，即以低能量的物理相关限制准确地解决该模型。我将对实验性可观察物进行预测，并将技术推广到其他分析应用。其次，我将在半数字方法“功能重新归一化组”（FRG）的上下文中探索“相互作用驱动的缩放”，该方法也没有人工小参数。此外，我计划了一个最近开发的称为“ Multiloop FRG”的扩展程序的应用，该扩展名特别适合以公正的方式了解竞争的订购趋势。因此，通过将其应用于已知可以托管QPT的简化电子模型，我希望能够更好地理解非常规超导性。