  We study a quantum dot coupled to two edge states of a quantum spin Hall
insulator through electron tunnelings in the presence of a Rashba spin-orbital
interaction induced by an external electric field. We show that if the electron
interactions on the edge states are repulsive, there are two possible phases,
depending on the Luttinger liquid parameter $K$. For $1/2<K<1$, the
low-temperature physics is controlled by a previously identified two-channel
Kondo fixed point. For the edge states with even stronger repulsive
interactions, i.e. $1/4<K<1/2$, the system reaches another phase at low
temperatures, described by a new two-channel Kondo fixed point. This phase is
separated from the original one by a continuous phase transition upon varying
the value of $K$ through the external electric field. The corresponding
critical point is described by a free Dirac fermion backscattered by a local
potential. We investigate the low-temperature properties associated with this
new fixed point and also discuss the scaling behaviors of the system at the
critical point.
