RKKY interaction and two Kondo impurities: the complete phase diagram

Since the seminal papers of Jones and Varma [1,2], the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction Y between two Kondo impurities is conventionally modeled by a direct Heisenberg coupling term J_H. It gives rise to a quantum phase transition between the Kondo and the RKKY phases in the two-impurity Kondo model for arbitrarily large Kondo couplings, even in the presence of charge fluctuations. However, the significance of this result is still controversial. Firstly, the transition is extremely fragile to particle-hole asymmetry, smearing the critical point into a crossover in its presence [3,4]. This has lead to the common belief that it cannot be achieved in a realistic experimental 2-impurity system and has made its relevance for lattice models debatable. Moreover, in the model Y and the Kondo exchange J_K are considered independent, although Y is genuinely generated from J_K, and the Kondo temperature depends on Y, as has been shown experimentally [5] and theoretically [6].

Recently, it has been shown that the quantum phase transition can be restored for weaker particle-hole symmetry by parameter fine-tuning [7]. We revisit the problem to show, by numerical renormalization group calculations, that for the simple model with RKKY interaction genuinely induced by Kondo couplings and the inter-host interaction, either two phase transitions occur or none, depending on the Kondo coupling strength. Each transition corresponds to destruction of effective quasi-particles by the relevant spin exchange and occurs at spin exchange strength of the order of the quasi-particle bandwidth. Similar results are expected for all quantum phase transitions induced by indirect interactions. Furthermore, we confirm earlier perturbative prediction [6] of the Kondo breakdown at a finite Kondo temperature T_K, with T_K defined as the scale where departure from high-energy local-moment regime happens, while strong-coupling or RKKY fixed points are achieved at arbitrarily small energy scales close to the transition. These findings may be relevant for heavy-fermion materials via DMFT mapping. We also propose experimental setup with 2-impurity system involving flat-band Moiré material for direct test of our predictions.

[1] B. A. Jones and C. M. Varma, Phys. Rev. Lett. 58, 843 (1987).
[2] B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev. Lett. 61, 125 (1988).
[3] R. M. Fye, Phys. Rev. Lett. 72, 916 (1994).
[4] I. Affleck, A. W. W. Ludwig, and B. A. Jones, Phys. Rev. B 52, 9528 (1995).
[5] J. Bork et al., Nat. Phys. 7, 901 (2011).
[6] A. Nejati, K. Ballmann, J. Kroha, Phys. Rev. Lett. 118, 117204 (2017).
[7] F. Eickhoff, B. Lechtenberg, F. Anders, Phys. Rev. B 98, 115103 (2018).
[8] K. P. Wójcik, J. Kroha, arXiv:2106.07519 (2021).