  The Jordan-Wigner transformation is applied to study the ground state
properties and dimerization transition in the $J_1-J_2$ $XXZ$ chain. We
consider different solutions of the mean-field approximation for the
transformed Hamiltonian. Ground state energy and the static structure factor
are compared with complementary exact diagonalization and good agreement is
found near the limit of the Majumdar-Ghosh model. Furthermore, the ground state
phase diagram is discussed within the mean-field theory. In particular, we show
that an incommensurate ground state is absent for large $J_2$ in a fully
self-consistent mean-field analysis.
