Competing order parameters and flows into phases with broken symmetries

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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.