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

Natural Orbitals Renormalization Group Approach to the Two-Impurity Kondo Critical Point