Surprises in the phase diagram of two impurities on flat-band hosts coupled by genuine RKKY interaction

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Since the seminal papers of Jones and Varma [1,2], the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction Y between two Kondo impurities is often modeled by a Heisenberg coupling term JH. 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 (the relevant phase diagram, obtained with a numerical renormalization group technique is shown in Figure 1 for a comparison with further results). However, the significance of this result is still controversial. Firstly, the transition is 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 realized in a realistic 2-impurity system and has made its relevance for lattice models debatable. Moreover, in this model the Y and the Kondo exchange JK are considered independent, although Y is genuinely generated from JK, 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 by considering a geometry of two impurities, where each one is coupled to a different host as in [5], and the RKKY interaction is induced solely by JK and an inter-host exchange coupling JY. We show by numerical renormalization group (NRG) calculations that, for a properly symmetric case, this causes a transition, if the Kondo coupling is not too strong. Moreover, another phase transition occurs then at very strong JY (of the order of the bandwidth D), which is not present in the Jones-Varma model [1,2], and drives the system to yet another phase, with non-universal impurity spectral density. The two phase transition lines meet for increasing JK and, hence, are replaced by a single crossover above a critical value of JK. The corresponding phase diagram is presented in Figure 2.The phase diagram can be understood by analyzing the relevant quasiparticle pictures. In the Kondo regime, the relevant quasi-particles are spatially somewhat extended objects, and their effective bandwidth is given by the width of the Abrikosov-Suhl resonance, which is of the order of the Kondo temperature TK. Therefore, the first transition (as recognized by Jones and Varma [2]) happens at the RKKY interaction strength of the order of TK. In the RKKY phase the Kondo quasiparticles are destroyed and the conduction electron states alone comprise the relevant quasiparticles. The second phase transition then happens when the inter-host coupling becomes of the order of the bandwidth D. For inter-host couplings exceeding the second critical value, the band electrons become irrelevant again and the effective quasiparticles (which we call Heisenberg quasi-particles) are somewhat analogous to the Kondo quasi-particles, in that the two hosts mutually spin-screen each other. The latter observation also explains why a continuous crossover is possible between the Kondo phase and the Heisenberg one.The two transitions present for genuine RKKY interaction can be recognized in the NRG results as abrupt changes of the local density of states of electrons localized at impurities and of conduction band electrons at the impurity sites, as well as from the divergence of corresponding staggered magnetic susceptibilities. This may make them detectable despite the fact that all static spin-spin correlation functions are continuous through both transitions (like in the Jones-Varma model). As the inter-host exchange must be of the order of the width of the conduction band for the second transition to be observed, the phenomenon seems relevant to for narrow-bandwidth materials, such as twisted bilayer graphene or other flat-band Moirés [8]. **