  We consider the interacting helical liquid system at the one-dimensional edge
of a two-dimensional topological insulator, coupled to an external magnetic
field and s-wave superconductor and map it to an XYZ spin chain system. This
model undergoes quantum Berezinskii-Kosterlitz-Thouless (BKT) transition with
two limiting conditions. We derive the renormalization group (RG) equations
explicitly and also present the flow lines behavior. We also present the
behavior of RG flow lines based on the exact solution. We observe that the
physics of Majorana fermion zero modes and the gaped Ising-ferromagnetic phase,
which appears in a different context. We observe that the evidence of gapless
helical Luttinger liquid phase as a common non-topological quantum phase for
both quantum BKT transitions. We explain analytically and physically that there
is no Majorana-Ising transition. In the presence of chemical potential, the
system shows the commensurate to incommensurate transition.
