Analysis of the magnetic translation group and investigation of a one-dimensional topological model

The periodicity of a space lattice in presence of a uniform magnetic eld is preserved. During this thesis, we will study a set of modi ed translation operators which commute with the e ective Hamiltonian of an electron in the lattice. Group theory helps us to construct matrix representations of the modi ed translation operators. These operators form ray groups. Using group projection operators, we will nd partner functions for constructed irreducible representation in order to obtain a relation which corresponds to Bloch function in a periodic lattice and is named as Bloch-type function. By multiplying a phase factor to modi ed translation operators, they will be extended to a new set of operators called magnetic translation operators so that they form a full group rather than a ray group. In a similar procedure, we will investigate displacement operators in phase space coordinate to form a full group of them. In another study, we will introduce a one dimensional model derived from Creutz model, called shifted Creutz model, in which a gap closure appears in its ground state band structure leading to timereversal symmetry breaking and subsequently giving rise to a topological phase transition. Adopting spin-orbit coupling to our model, generates a time-reversal symmetric pair of states with two-fold degeneracy. A topological investigation will be carried on both models by analyzing the band structures, phase diagram, edge states, symmetries in the models, and calculating the winding number.