The development of two-dimensional materials in order to realize certain valuable electronic properties is a rapidly ongoing process. In this context, Dirac materials which exhibit low-energy Dirac-like excitations are of special interest, with graphene being the most famous representative. In order to understand existing phenomena and to extend application possibilities, new phases are constantly considered, leading to a rather rich phase diagram containing distinct ordered phases of matter. Here the competition between those phases with regard to the most preferred realization becomes important.In this thesis we address the semimetal-insulator phase transition where the insulating phase can either be realized in terms of an antiferromagnetic phase or a staggered charge density configuration. To this end we establish the Dyson-Schwinger formalism on the hexagonal lattice without the common low-energy approximations, taking the whole band structure of graphene into account. These collective phenomena are then studied under the influence of an extended Hubbard interaction within several truncation schemes. Furthermore, the impact of varying temperature and chemical potential on the phase transition is investigated. Additionally we also address the corresponding critical exponents characterizing the phase transition in terms of specific universality classes.
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