  The problem of two magnetic impurities in a normal metal exposes the two
opposite tendencies in the formation of a singlet ground state, driven
respectively by the single-ion Kondo effect with conduction electrons to screen
impurity spins or the Ruderman-Kittel-Kasuya-Yosida interaction between the two
impurities to directly form impurity spin singlet. However, whether the
competition between these two tendencies can lead to a quantum critical point
has been debated over more than two decades. Here, we study this problem by
applying the newly proposed natural orbitals renormalization group method to a
lattice version of the two-impurity Kondo model with a direct exchange $K$
between the two impurity spins. The method allows for unbiased accessing the
ground state wave functions and low-lying excitations for sufficiently large
system sizes. We demonstrate the existence of a quantum critical point,
characterized by the power-law divergence of impurity staggered susceptibility
with critical exponent $\gamma = 0.60(1)$, on the antiferromagnetic side of $K$
when the interimpurity distance $R$ is even lattice spacing, while a crossover
behavior is recovered when $R$ is odd lattice spacing. These results have
ultimately resolved the long-standing discrepancy between the numerical
renormalization group and quantum Monte Carlo studies, confirming a link of
this two-impurity Kondo critical point to a hidden particle-hole symmetry
predicted by the local Fermi liquid theory.
