The Kane and Mele model is two copies of the Haldane model such that the spin up electron exhibits a chiral integer quantum Hall Effect while the spin down electron exhibits an anti-chiral integer quantum Hall effect.
A relativistic version of the quantum spin Hall effect was introduced in the 1990s for the numerical simulation of chiral gauge theories;[3][4] the simplest example consisting of a parity and time reversal symmetric U(1) gauge theory with bulk fermions of opposite sign mass, a massless Dirac surface mode, and bulk currents that carry chirality but not charge (the spin Hall current analogue).
Independently, a quantum spin Hall model was proposed by Andrei Bernevig and Shoucheng Zhang[5] in an intricate strain architecture which engineers, due to spin-orbit coupling, a magnetic field pointing upwards for spin-up electrons and a magnetic field pointing downwards for spin-down electrons.
Further stability studies of the edge liquid through which conduction takes place in the quantum spin Hall state proved, both analytically and numerically that the non-trivial state is robust to both interactions and extra spin-orbit coupling terms that mix spin-up and spin-down electrons.
Since graphene has extremely weak spin-orbit coupling, it is very unlikely to support a quantum spin Hall state at temperatures achievable with today's technologies.
Two-dimensional topological insulators (also known as the quantum spin Hall insulators) with one-dimensional helical edge states were predicted in 2006 by Bernevig, Hughes and Zhang to occur in quantum wells (very thin layers) of mercury telluride sandwiched between cadmium telluride,[7] and were observed in 2007.
When the sheet of HgTe in between the CdTe is thin, the system behaves like an ordinary insulator and does not conduct when the Fermi level resides in the band-gap.
When the sheet of HgTe is varied and made thicker (this requires the fabrication of separate quantum wells), an interesting phenomenon happens.
In the gap closing and re-opening process, two edge states are brought out from the bulk and cross the bulk-gap.
A magnetic field should destroy the quantum spin Hall state by breaking time-reversal invariance and allowing spin-up spin-down electron scattering processes at the edge.
All these predictions have been experimentally verified in an experiment [9] performed in the Molenkamp labs at Universität Würzburg in Germany.