Thermodynamic quantum critical behavior of the Kondo necklace model

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We obtain the phase diagram and thermodynamic behavior of the Kondo necklace model for arbitrary dimensions $d$ using a representation for the localized and conduction electrons in terms of local Kondo singlet and triplet operators. A decoupling scheme on the double time Green's functions yields the dispersion relation for the excitations of the system. We show that in $d\ensuremath{\geqslant}3$, there is an antiferromagnetically ordered state at finite temperatures terminating at a quantum critical point (QCP). In two dimensions, long-range magnetic order occurs only at $T=0$. The line of N\'eel transitions for $d>2$ varies with the distance to the quantum critical point QCP $\ensuremath{\mid}g\ensuremath{\mid}$ as ${T}_{N}\ensuremath{\propto}{\ensuremath{\mid}g\ensuremath{\mid}}^{\ensuremath{\psi}}$, where the shift exponent $\ensuremath{\psi}=1∕(d\ensuremath{-}1)$. In the paramagnetic side of the phase diagram, the spin gap behaves as $\ensuremath{\Delta}\ensuremath{\approx}\sqrt{\ensuremath{\mid}g\ensuremath{\mid}}$ for $d\ensuremath{\geqslant}3$, consistent with the value $z=1$ found for the dynamical critical exponent. We also find in this region a power law temperature dependence in the specific heat for ${k}_{B}T⪢\ensuremath{\Delta}$ and along the non-Fermi-liquid trajectory. For ${k}_{B}T⪡\ensuremath{\Delta}$, in the so-called Kondo spin liquid phase, the thermodynamic behavior is dominated by an exponential temperature dependence.