A Pedagogic Review on Designing Model Topological Insulators

Abstract

Following the centuries old concept of the quantization of fluxthrough a Gaussian curvature (Euler characteristic) and its successivedispersal into various condensed matter properties such as quantum Halleffect, and topological invariants, we can establish a simple and fairlyuniversal understanding of various modern topological insulators (TIs).Formation of a periodic lattice (which is a non-trivial Gaussian curvature) of‘cyclotron orbits’ with applied magnetic field, or ‘chiral orbits’ with fictitious‘momentum space magnetic field’ (Berry curvature) guarantees its fluxquantization, and thus integer quantum Hall (IQH), and quantum spin-Hall(QSH) insulators, respectively, occur. The bulk-boundary correspondence associated with all classes of TIs dictates that some sort of pumping orpolarization of a ‘quantity’ at the boundary must be associated with theflux quantization or topological invariant in the bulk. Unlike charge or spin accumulations at the edge for IQH and QSH states, the time-reversal (TR) invariant Z2 TI class pumps a mathematical quantity called ‘TR polarization’to the surface. This requires that the valence electron’s wavefunction (say,ψ ↑(k)) switches to its TR conjugate ψ↓ ( †(-k) ) odd number of times in halfof the Brillouin zone. These two universal features can be considered as‘targets’ to design and predict various TIs. For example, we demonstrate that when two adjacent atomic chains or layers are assembled with opposites pin-orbit coupling (SOC), serving as the TR partner to each other, the system naturally becomes a Z2 TI. This review delivers a holistic overview on various concepts, computational schemes, and engineering principles of TIs.