  We determine the phase diagrams of anisotropic Kitaev-Heisenberg models on
the honeycomb lattice using parton mean-field theories based on different
Majorana fermion representations of the $S=1/2$ spin operators. Firstly, we use
a two-dimensional Jordan-Wigner transformation (JWT) involving a semi-infinite
snake string operator. In order to ensure that the fermionized Hamiltonian
remains local we consider the limit of extreme Ising exchange anisotropy in the
Heisenberg sector. Secondly, we use the conventional Kitaev representation in
terms of four Majorana fermions subject to local constraints, which we enforce
through Lagrange multipliers. For both representations we self-consistently
decouple the interaction terms in the bond and magnetization channels and
determine the phase diagrams as a function of the anisotropy of the Kitaev
couplings and the relative strength of the Ising exchange. While both
mean-field theories produce identical phase boundaries for the topological
phase transition between the gapless and gapped Kitaev quantum spin liquids,
the JWT fails to correctly describe the the magnetic instability and
finite-temperature behavior. Our results show that the magnetic phase
transition is first order at low temperatures but becomes continuous above a
certain temperature. At this energy scale we also observe a finite temperature
crossover on the quantum spin-liquid side, from a fractionalized paramagnet at
low temperatures, in which gapped flux excitations are frozen out, to a
conventional paramagnet at high temperatures.
