  We study a quantum point contact in a quantum spin Hall insulator. It has
recently been shown that the Luttinger liquid theory of such a structure maps
to the theory of a weak link in a Luttinger liquid with spin with Luttinger
liquid parameters g_\rho = 1/g_\sigma = g < 1. We show that for 1/2<g<1, the
pinch-off of the point contact as a function of gate voltage is controlled by a
novel quantum critical point, related to a nontrivial intermediate fixed point
found previously in the Luttinger liquid model. We predict that the dependence
of the conductance on gate voltage and temperature near the pinch-off
transition collapses onto a universal curve described by a crossover scaling
function. We compute the conductance, the critical exponents and the scaling
function in solvable limits, which include g=1-\epsilon, g=1/2+\epsilon and
g=1/\sqrt{3}. These results, along with a general scaling analysis provide an
overall picture of the critical behavior as a function of g. In addition, we
analyze the structure of the four terminal conductance of the point contact in
the weak tunneling and weak backscattering limits. We find that different
components of the conductance can have different temperature dependence. We
identify a skew conductance G_{XY}, which we predict vanishes as T^\gamma with
\gamma\ge 2. This behavior is a direct consequence of the unique edge state
structure of the quantum spin Hall insulator. Finally, we show that for g<1/2
the presence of spin non conserving spin orbit interactions leads to a novel
time reversal symmetry breaking insulating phase. In this phase, the transport
is carried by spinless chargons and chargeless spinons. These lead to
nontrivial correlations in the low frequency shot noise. Implications for
experiments on HgCdTe quantum well structures will be discussed.
