Properties And Applications Of Low-dimensional Topological Insulators And Topological Superconductors

Topological insulators and topological superconductors (or superfluids) are two new classes of topological phases, they share similar properties with other topological phases, like integer quantum Hall effect. With periodic boundary condition, both the spectra of topological insulators and topological superconductors have a non-vanishing energy gap, as a result, they can be similarly classified into different topological classes according to the symmetries of their hamiltonians. When the topological insulators and topological superconductors are in their topologically nontrivial phases, robust gapless edge states will show up at their open boundaries. The robust gapless edge state is one of the most prominent properties of topological insulators, because this makes the surface of the insulators in fact metallic. For the topological superconductors, what most interests us is the existence of Majorana zero modes located at defects like vortex, or the boundary of one dimensional systems. This thesis is organized as follows:In chapter1, I give an introduction to the concepts of topological order, topo-logical insulator and topological superconductor. The periodic table classifying the topological insulators and topological superconductors is also given explicitly.Chapter2and chapter3are devoted to discuss our works on topoiogical insu-lators. Concretely, in chapter2, we introduce spin-orbit coupling to the well-known Su-Schrieffer-Heeger (SSH) model and study the topological properties of the model. In the absence of spin-orbit coupling, the original SSH model has two topologically distinct insulating phases, one has no zero-energy bound states and the other one has two zero-energy bound states at each end of the one dimensional system. However, due to the existence of soliton, neither of the two phases are truly insulating phases, on the contrary, the system is generally metallic. The main result of the introduction of spin-orbit coupling is the emergence of a new topological phase in which only a single zero-energy bound state is located at each end of the system. In this new topological phase, the interesting flat band of nontrivial topology forms in a wide range of param-eters, and furthermore, there is no soliton, which suggests that this new phase is a truly topological insulating phase. In chapter3, we study the Floquet approach to change the topological properties of time-reversal symmetric cold-atomic systems. We find that the general method in condensed matter systems that using a time-periodic elec-tromagnetic field is incapable of changing the topological properties of time-reversal symmetric cold-atomic systems. For such systems, we find that periodically varying the potential of optical lattice provides a very simple and effective way to induce topo-logical phase transition. We also check that this simple method can also be applied to time-reversal-symmetry-breaking systems.Chapter4and chapter5are devoted to discuss our works on topological super-fluids and topological superconductors. Concretely, in chapter4, we study two mod-els that can realize time-reversal symmetric topological superfluids and time-reversal-symmetry-breaking topological superfluids, respectively. For the time-reversal sym-metric case, we give a detail analysis of the topological criteria and show the phase diagram obtained by self-consistently solving the order parameter of the superfluid. For the time-reversal-symmetry-breaking case, the non-superconducting model is al-ready very interesting since it can host chiral edge states. With the introduction of a simple yet realizable attractive Hubbard interaction, we find that by pairing the chiral edge states, topological superfluidity will naturally emerge. What’s more, when the chemical potential intersects with the chiral edge modes, we find that the parity of the number of edge modes directly determines the topological property of the superfluid. We have also studied the effect of the appearance of a harmonic trap, what we find is that when the middle region is topologically nontrivial, the Majorana zero modes will always show up, but they are no longer localized at the boundary of the system. In chapter5, by taking a spinless p-wave superconductor as an example and using the Blonder-Tinkham-Klapwijk approach, we show that the zero-bias conductance of a normal metal/p-wave superconductors can directly reflect the topological property of the superconductor. When the p-wave superconductor is topologically nontrivial, it is found that the zero-bias conductance takes a quantized value2(c2)/h, but when it is topo-logically trivial, the zero-bias conductance is found to vanish, which suggests that the zero-bias conductance can be used to determine the topological properties of a super-conductor. At the second part of this chapter, we show how to make the application of a time-reversal symmetric topological superconductor to measure the spin-polarization of ferromagnets which play a fundamental role in spintronics. Concretely, we find that the zero-bias conductance of the junction composed by a ferromagnet and time-reversal symmetric topological superconductor with un-spin-polarized pairing type is of topo-logical nature in the sense that it only depends on the parameters of the ferromagnet. Compared to the traditional method by adopting s-wave superconductors, as now the zero-bias conductance is independent of the tunneling barrier and parameters of the superconductor, our method provides a simper and more direct way to determine the spin-polarization.In chapter6,1give a brief conclusion and outlook of this thesis.