One-dimensional chiral topological insulators in ladder models

The topological insulator state describes a quantum state of matter that hosts topologically protected conductive edge modes, similar to the quantum spin Hall state but without the necessary applied fields. The coupling of topology to condensed matter theory in the context of topological insulators has produced the periodic table of topological insulators and superconductors. The ever popular SSH chain has become the prototypical model for describing simple topological properties in one-dimension. It is widely known that this model, in its modern form, hosts symmetry protected end states that are at zero energy. This model has supported mountains of research. In this thesis we will use two deformed SSH model chains to constuct ladder models that are topological insulators in each of the chiral universality classes (AIII, BDI, CII, DIII, and CI). We systematically construct these models from general forms to specific forms examining their energy and wavefunction spectrums, leading easily to the conclusion of their topological nature. These constructions are motivated by the finding that the winding number has a sign ambiguity. This ambiguity leads to two forms of the chiral symmetry operator and subsequently two forms of the ladder, one adhering to the BDI and CII universality classes, and one for the DIII and CI classes. We are able to analytical derive expressions for the edge states in each chiral class ladder model and demonstrate the symmetry properties of each are encoded in these states. Additionally we show that, as a consequence of the sign ambiguity, for a weak interchain coupling, the winding numbers of the individual chains can be added leading to an index of 2, in the case of the BDI and CII class models, or subtracted giving an index of 0 in the case of the DIII and CI models. The conclusions and properties from this section of work is completely general and applicable to other chiral models. In the final section of research we show via diagrammatic arguments and mean field theory the existence of a Z2 symmetry breaking bond density wave ground state in an SSH-like model with density-density interactions and a reduced filling fraction. The reduced filling fraction stabilizes a topological ground state where interactions would normally not permit one to exist. We further demonstrate this ground state occurs as a result of mean field rather than strong correlation effects. The ground state model turns out to be the noninteracting SSH4 model which we show has quantized Zak phases of individual bands confirming the topological nature of the model in a reduced filling fraction regime. Publication: P. Matveeva, T. Hewitt, D. Liu, K. Reddy, D. Gutman, and S. T. Carr, ‘One dimensional noninteracting topological insulators with chiral symmetry’, Phys. Rev. B., vol. 107, no. 7, Feb. 2023.