Modeling Axion Insulators

The Chern-Simons magnetoelectric coupling can be used to provide a Z₂ topological classification of 3D insulators having either time-reversal or inversion symmetry. When time-reversal is present, the invariant distinguishes trivial from strong topological insulators; when inversion is present, the topological phase is called an axion insulator. Here we describe two tight-binding models that provide insight into the topological properties of axion insulators and their surfaces. The first one starts from a simple k-space Hamiltonian that is transformed to real space and manipulated to obtain an axion insulator. The second model is more physically motivated, being based on previous work intended to model the topological properties of pyrochlore iridates, a class of materials that have been of considerable interest due to an interplay between electronic and spin-orbit interactions. We use these models to clarify the conditions under which the surface can be gapped and, when it is gapped, how the sign of the half-integer quantum Hall effect at the surface is determined.