  The search for topologically non-trivial states of matter has become an
important goal for condensed matter physics. Recently, a new class of
topological insulators has been proposed. These topological insulators have an
insulating gap in the bulk, but have topologically protected edge states due to
the time reversal symmetry. In two dimensions the helical edge states give rise
to the quantum spin Hall (QSH) effect, in the absence of any external magnetic
field. Here we review a recent theory which predicts that the QSH state can be
realized in HgTe/CdTe semiconductor quantum wells. By varying the thickness of
the quantum well, the band structure changes from a normal to an "inverted"
type at a critical thickness $d_c$. We present an analytical solution of the
helical edge states and explicitly demonstrate their topological stability. We
also review the recent experimental observation of the QSH state in
HgTe/(Hg,Cd)Te quantum wells. We review both the fabrication of the sample and
the experimental setup. For thin quantum wells with well width $d_{QW}< 6.3$
nm, the insulating regime shows the conventional behavior of vanishingly small
conductance at low temperature. However, for thicker quantum wells ($d_{QW}>
6.3$ nm), the nominally insulating regime shows a plateau of residual
conductance close to $2e^2/h$. The residual conductance is independent of the
sample width, indicating that it is caused by edge states. Furthermore, the
residual conductance is destroyed by a small external magnetic field. The
quantum phase transition at the critical thickness, $d_c= 6.3$ nm, is also
independently determined from the occurrence of a magnetic field induced
insulator to metal transition.
