Competing order parameters and flows into phases with broken symmetries

对竞争有序趋势的理解仍然是现代凝聚态物理学中最相关的问题之一。在这个项目中，我们使用函数RG来研究这种竞争，包括上述和二维相关费米子模型的对称破缺相，总体目标是确定相边界，序参数对称性和大小，并理解低能行为。主要目的是对对称破缺尺度上下的二维Hubbard模型中RG流中磁性、超导、正向和umklapp散射项的相互作用进行控制分析。为此，我们想比较和结合我们小组在过去三年中获得的关于费米子和玻色子功能RG流的技术和见解，以便跟踪流到最低尺度，从而确定接近半填充的相图。其中涉及的重要问题包括：(1)确定费米子相互作用顶点的哪些部分（对应于交换玻色子）需要在不同的尺度上被包括在内；(2)跟踪自能、汤川型顶点函数和四费米子项的动量和频率依赖关系，以及对称破缺区局部势的函数依赖关系；(3)确保流动函数渐近地满足Ward恒等式。此外，我们将使用我们的方法来更深入地研究现代相关电子系统的其他例子，例如非常规超导体或拓扑电子态。

The understanding of competing ordering tendencies remains one of the most relevant questions in modern condensed matter physics. In this project we use the functional RG to study this competition, both above and in the symmetry-broken phase of two-dimensional correlated fermion models, with the overall goal of determining phase boundaries, order parameter symmetries and magnitudes, and understanding the low-energy behaviour.A major aim is a controlled analysis of the interplay of magnetic, superconducting, forward and umklapp scattering terms in the RG flow for the two-dimensional Hubbard model both above and below the symmetry-breaking scale. To this end, we want to compare and combine the techniques and insights obtained in our group on fermionic and bosonic functional RG flows during the past three years, in order to follow the flow to the lowest scales and thus determine the phase diagram close to half-filling.Important issues involved in this include (1) determining which parts of the fermionic interaction vertex (corresponding to exchange bosons) need to be included at different scales, (2) tracking the momentum and frequency dependence of self-energies, Yukawa-type vertex functions and four-fermion terms, as well as the functional dependence of local potentials in the symmetry-broken regime, and (3) ensuring that the flowing functions asymptotically fulfill the Ward identities. Furthermore, we will use our method to examine other examples of modern correlated electron system, such as unconventional superconductors or topological electron states in more depth.