Jordan-Wigner approach to the frustrated spin one-half XXZ chain

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.