Thermodynamic quantum critical behavior of the anisotropic Kondo necklace model

The Ising-like anisotropy parameter δ in the Kondo necklace model is analyzed using the bond-operator method at zero and finite temperatures for arbitrary d dimensions. A decoupling scheme on the double time Green's functions is used to find the dispersion relation for the excitations of the system. At zero temperature and in the paramagnetic side of the phase diagram, we determine the spin gap exponent νz≈0.5 in three dimensions and anisotropy between 0?δ?1, a result consistent with the dynamic exponent z=1 for the Gaussian character of the bond-operator treatment. On the other hand, in the antiferromagnetic phase at low but finite temperatures, the line of Neel transitions is calculated for δ?1. For d>2 it is only re-normalized by the anisotropy parameter and varies with the distance to the quantum critical point (QCP) |g| as, T_N∝|g|^ψ where the shift exponent ψ=1/(d-1). Nevertheless, in two dimensions, a long-range magnetic order occurs only at T=0 for any δ?1. In the paramagnetic phase, we also find a power law temperature dependence on the specific heat at the quantum critical trajectoryJ/t=(J/t)_c, T→0. It behaves as C_V∝T^d for δ?1 and ≈1, in concordance with the scaling theory for z=1.