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Many important materials such as high-temperature superconductors and heavy-fermion compounds contain electrons that interact so strongly with each other that conventional methods fail to explain their behaviour. In this thesis, we develop a new theoretical method to study such systems by relating a system of interacting electrons moving on a lattice to a much simpler “auxiliary” quantum impurity model, where all the interaction is focused at a single point and the rest of the system is non-interacting. Instead of solving the full many-body problem directly, we solve the impurity model and then reconstruct the behaviour of the entire lattice from it. This reconstruction works through a mathematical “tiling” procedure that reproduces the periodic structure of the lattice through a many-electron version of Bloch's theorem, and generates momentum-dependent physical quantities. Using this framework, we study paradigmatic problems such as the Mott transition in two dimensions and quantum critical behaviour in heavy-fermion systems. Our investigations reveal how, when pushed to the brink of a transition, electrons morph into new emergent particles and lead to drastically new behaviour in metals such as pseudogap and non-Fermi liquid physics. |