Study of classical and quantum phase transitions on non-Euclidean geometries in higher dimensions

The investigation of the behaviour of both classical and quantum systems on non- Euclidean surfaces near the phase transition point represents an interesting research area of the modern physics. However, due to the specific nature of the hyperbolic geometry, there have been no analytical solutions available so far and the potential of analytic and standard numerical methods is strongly limited. The task of finding an appropriate approach to analyze the fermionic models on hyperbolic lattices in the thermodynamic limit still remains an open question. In case of classical spin systems, a generalization of the Corner Transfer Matrix Renormalization Group algorithm has been developed and successfully applied to spin models on infinitely many regular hyperbolic lattices. In this work, we extend these studies to specific types of lattices. We also conclude that the hyperbolic geometry induces mean-field behaviour of all spin models at phase transitions. It is important to mention that no suitable algorithms for numerical analysis of ground-states of quantum systems in similar conditions have been implemented yet. In this thesis we offer a particular solution of the problem by proposing a variational numerical algorithm Tensor Product Variational Formulation, which assumes a quantum ground-state written in the form of a low-dimensional uniform tensor product state. We apply the Tensor Product Variational Formulation to three typical quantum models on a variety of regular hyperbolic lattices. Again, as in the case of classical spin systems, we conjecture the identical adherence to the mean-field-like universality class irrespective of the original model. The main outcomes of this thesis are the following: (1) We propose an algorithm for calculation and classification of the thermodynamic properties of the Ising model on triangular-tiled hyperbolic lattices. In addition, we investigate the origin of the mean-field universality on a series of weakly curved lattices. (2) We develop the Tensor Product Variational Formulation algorithm for the numerical analysis of the ground-state of the quantum systems on the hyperbolic lattices. (3) We study quantum phase transition phenomena for the three selected spin models on various types of the hyperbolic lattices including the Bethe lattice. (Author)