Electronic correlations result in emergent collective phenomena. Our goal in this grant period is to pur- sue our numerical investigations of model systems where the emergent phenomena is of topological nature and competes or is intertwined with other orders. Correlation induced topological phenom- ena can be found in a variety of model systems. In the framework of heavy fermion systems the interplay of correlation and spin-orbit coupling can lead to an emergent band structure that carries a topological index. In this funding period, we will concentrate on so called Kondo semi-metals with emergent Dirac particles. The electron-phonon problem, as realized by the so called Su-Schrieffer- Heeger (SSH) model on a two dimensional lattice, is extremely challenging and to date the phase diagram has remained elusive. In the first grant period we have developed a quantum Monte Carlo (QMC) method amenable to simulate this model and we are now in a position to map out the phase diagram. Aside from the competition between superconductivity and bond-density wave states, we foresee exotic quantum phase transitions as well as topologically ordered phases. Strong correla- tions in Dirac systems can lead to a variety of quantum phase transitions as well as to the breakdown of the Fermi liquid. Using so called SLAC fermions – enabling us to simulate a single Dirac cone sub- ject to a long range Coulomb repulsion – we will further investigate the fermi velocity renormalization, transport properties as well as the Fermi surface anisotropy renormalization. The latter is of particular interest given the relationship between the half-filled Landau level and Dirac fermions subject to the Coulomb repulsion. For all the above research subjects, we will use unbiased numerical approaches based on various auxiliary field QMC algorithms. We will embed all the code development in our Algorithms for Lattice Fermions (ALF) program package.