Two-site equation of motion technique - a Hubbard model study One of the main problems of understanding strongly correlated systems is that there is no universally accepted sufficiently simple reference system available for their description. The main workhorse of condensed matter physics, namely independent fermions, is not a good place to start. A better starting point is operators that are closer to creating the actual excitations in the interacting system. These can be identified by studying the equations of motion (EOM) for some chosen class of operators. In such a scheme there are a number of issues that must be dealt with: 1) Which class of operators should be used in the construction? The conventional approach is to use intuition from the physics to select the operators. 2) The next step is to identify or generate the operator dynamics. Ideally this dynamics is a reasonable approximation to reality. In practice this implies that the EOM have to be closed in a way that keeps the theory physical, for example keeping spectral weights positive. Also this step is not unique. 3) Once 1) and 2) are settled the EOM have to be solved and the averages that are necessary to construct the Green’s functions must be computable from within the theory itself. In our work, we suggest to use all operators that affect up to a pair of sites in the construction, making the scheme as unbiased as possible. This also has the advantage that the necessary averages can be calculated within the theory. We illustrate our scheme by an application to the Hubbard model. EOM technique introduction: Avella & Mancini, http://arxiv.org/abs/1112.1103 Mott physics introduction: Phillips, http://arxiv.org/abs/1001.5270

Topological Superconductors and Majorana Fermions Topological superconductors (TSCs) are a newly discovered class of superconductors with features uniquely advantageous for quantum computing. They have recently raised an immense amount of attention due to the possibility of them hosting Majorana fermion quasiparticles at surfaces, vortices, and other defects. Approximately one can say that a Majorana fermion quasiparticle is half an electron, or more accurately, in a material with Majorana fermions the wave function of an electron has split up into two separate parts. This non-local property of two Majorana fermions can be used for exceptionally fault-tolerant quantum computing, where disorder and decoherence, currently severely hampering quantum computation, is automatically rendered unimportant. Our current work focuses on the presently most promising TSCs in superconducting hybrid structures of well-known spin-orbit coupled materials, as well as on discovering new and experimentally feasible TSCs in especially graphene- like materials. We have recently developed a computationally highly efficient framework to study superconductivity self- consistently in realistic lattice models. This allows us to access important microscopic details in general hybrid structures at an essentially unprecedented level. In this talk I will show some of our recent results on vortices, single magnetic impurities, and magnetic impurity wire networks. General references to the field: J. Alicea, Rep. Prog. Phys. 75, 076501 (2012); Y. Ando and L. Fu, Annu. Rev. Condens. Matter Phys. 6, 361 (2015). Recent preprint: Bjornson and Black-Schaffer, arXiv:1605.00696.