  A simplified version of the symmetric Kondo lattice model, the Kondo necklace
model, is studied by using a representation of impurity and conduction electron
spins in terms of local Kondo singlet and triplet operators. Within a mean
field theory, a spin gap always appears in the spin triplet excitation spectrum
in 1D, leading to a Kondo spin liquid state for any finite values of coupling
strength $t/J$ (with $t$ as hopping and $J$ as exchange); in 2D and 3D cubic
lattices the spin gaps are found to vanish continuously around $(t/J)_c\approx
0.70$ and $(t/J)_c\approx 0.38$, respectively, where quantum phase transitions
occur and the Kondo spin liquid state changes into an antiferromagnetically
long-range ordered state. These results are in agreement with variational Monte
Carlo, higher-order series expansion, and recent quantum Monte Carlo
calculations for the symmetric Kondo lattice model
