Quantum spin Hall effect
The quantum spin Hall state is a state of matter proposed to exist in special, two-dimensional, semiconductors that have a quantized spin-Hall conductance and a vanishing charge-Hall conductance. The quantum spin Hall state of matter is the cousin of the integer quantum Hall state, and that does not require the application of a large magnetic field. The quantum spin Hall state does not break charge conservation symmetry and spin- conservation symmetry (in order to have well defined Hall conductances). The first proposal for the existence of a quantum spin Hall state was developed by Kane and Mele who adapted an earlier model for graphene by F. Duncan M. Haldane which exhibits an integer quantum Hall effect. 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; 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).
Overall the Kane-Mele model has a charge-Hall conductance of exactly zero but a spin-Hall conductance of exactly (in units of ). Independently, a quantum spin Hall model was proposed by Bernevig and Zhang 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. The main ingredient is the existence of spin-orbit coupling, which can be understood as a momentum-dependent magnetic field coupling to the spin of the electron.
Real experimental systems, however, are far from the idealized picture presented above in which spin-up and spin-down electrons are not coupled. A very important achievement was the realization that the quantum spin Hall state remains to be non-trivial even after the introduction of spin-up spin-down scattering, which destroys the quantum spin Hall effect. In a separate paper, Kane and Mele introduced a topological invariant which characterizes a state as trivial or non-trivial band insulator (regardless if the state exhibits or does not exhibit a quantum spin Hall effect). 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. Such a non-trivial state (exhibiting or not exhibiting a quantum spin Hall effect) is called a topological insulator, which is an example of symmetry-protected topological order protected by charge conservation symmetry and time reversal symmetry. (Note that the quantum spin Hall state is also a symmetry-protected topological state protected by charge conservation symmetry and spin- conservation symmetry. We do not need time reversal symmetry to protect quantum spin Hall state. Topological insulator and quantum spin Hall state are different symmetry-protected topological states. So Topological insulator and quantum spin Hall state are different states of matter.)
In HgTe quantum wells
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. A very realistic theoretical proposal for the existence of the quantum spin Hall state has been put forward in 1987 by Pankratov, Pakhomov and Volkov in Cadmium Telluride/Mercury Telluride/Cadmium Telluride (CdTe/HgTe/CdTe) quantum wells in which a thin (5-7 nanometers) sheet of HgTe is sandwiched between two sheets of CdTe, and subsequently experimentally realized  (In fact, the proposed quantum wells do not exhibit the quantum spin Hall effect and do not demonstrate the existence of the quantum spin Hall state. It should be regarded as topological insulators. So far quantum spin Hall effect has not been observed in experiments.) Different quantum wells of varying HgTe thickness can be built. 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. Due to the inverted band structure of HgTe, at some critical HgTe thickness, a Lifshitz transition occurs in which the system closes the bulk band gap to become a semi-metal, and then re-opens it to become a quantum spin Hall insulator.
In the gap closing and re-opening process, two edge states are brought out from the bulk and cross the bulk-gap. As such, when the Fermi level resides in the bulk gap, the conduction is dominated by the edge channels that cross the gap. The two-terminal conductance is in the quantum spin Hall state and zero in the normal insulating state. As the conduction is dominated by the edge channels, the value of the conductance should be insensitive to how wide the sample is. 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  performed in the Molenkamp labs at Universität Würzburg in Germany. (In fact, a magnetic field in z-direction does not destroy the quantum spin Hall state which has conserved spins. The fact that the quantization of the two-terminal conductance is destroyed by magnetic field suggests that the quantum well is not a quantum spin Hall state, but a topological insulator.)
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