Topological Insulators

Predicted by Kane & Mele (2005), and Fu and Kane (2006) as a class of 2D nonmagnetic spin Hall insulators whose Bloch states are topologically twisted. Occurs when the charge hall conductance is zero, but the spin hall conductance is non-zero and takes quantized values. Gapless surface states come in pairs guaranteed by Kramer’s degeneracy in the presence of TRS.

Kane-Mele Model

With spin–orbit coupling, graphene can support edge states forming a QSH insulator, protected by time-reversal symmetry.

3D TI’s with TRS

The identification of “weak” and “strong” topological insulators by Fu et. al (2007), and Moore and Balents (2007)

Tight-binding model

Fu-Kane-Mele model for 3D topological insulator on a diamond lattice $H_{FKM}={\sum }_{⟨ij⟩}{t}_{ij}{c}_i^{†}{c}_j+i\frac{4{\lambda }_{SO}}{a^2}{\sum }_{⟨⟨ij⟩⟩}{c}_i^{†}\sigma \cdot ({\mathbf{d}}_{ij}^1\times {\mathbf{d}}_{ij}^2)c_j$