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Abstract

This thesis is dedicated to provide physicists with new and improved techniques to examine phase diagrams and phase transitions. One the one hand, an analysis of the effects of different regularization schemes in functional renormalization group calculations is provided. Building on this knowledge, an investigation of the phase diagram of the Hubbard-Model on the square lattice is performed using the functional renormalization group. The calculation reveals leading instabilities in the d-wave superconducting and different antiferromagnetic channels. In the symmetry broken phases there are a changing Fermi surface geometry, coexistence phases of d-wave superconductivity and antiferromagnetism as well as a mutual tendency of superconductivity and antiferromagnetism to repel each other. On the other hand, a scheme to discover phase transitions using unsupervised artifcial neural networks is developed. Further, a method to interpret artifcial neural networks is introduced. These methods are applied to systems ranging from the two dimensional Ising Model to four dimensional SU(2) lattice gauge theory. They find the existence of different phases, calculate phase boundaries and derive the explicit formulas of the quantities by which the neural network distinguishes between phases. It turns out that these quantities are order parameters and other thermodynamic quantities.