  In heavy-fermion systems, the competition between the local Kondo physics and
intersite magnetic fluctuations results in unconventional quantum critical
phenomena which are frequently addressed within the Kondo lattice model (KLM).
Here we study this interplay in the SU($N$) symmetric generalization of the
two-dimensional half-filled KLM by quantum Monte Carlo simulations with $N$ up
to 8. While the long-range antiferromagnetic (AF) order in SU($N$) quantum spin
systems typically gives way to spin-singlet ground states with spontaneously
broken lattice symmetry, we find that the SU($N$) KLM is unique in that for
each finite $N$ its ground-state phase diagram hosts only two phases -- AF
order and the Kondo-screened phase. The absence of any intermediate phase
between the $N=2$ and large-$N$ cases establishes adiabatic correspondence
between both limits and confirms that the large-$N$ theory is a correct saddle
point of the KLM fermionic path integral and a good starting point to include
quantum fluctuations. In addition, we determine the evolution of the
single-particle gap, quasiparticle residue of the doped hole at momentum
$(\pi,\pi)$, and spin gap across the magnetic order-disorder transition. Our
results indicate that increasing $N$ modifies the behavior of the coherence
temperature: while it evolves smoothly across the magnetic transition at $N=2$
it develops an abrupt jump -- of up to an order of magnitude -- at larger but
finite $N$. We discuss the magnetic order-disorder transition from a
quantum-field-theoretic perspective and comment on implications of our findings
for the interpretation of experiments on quantum critical heavy-fermion
compounds.
