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
