Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

RKKY interaction and two Kondo impurities: the complete phase diagram

Krzysztof P. Wójcik, Johann Kroha

Published Jun 14, 2021• 0 citations
OpenAlex

Abstract

Since Jones-Varma seminal works the indirect RKKY interaction is usually modeled as independently tunable Heisenberg spin exchange. For such a model in particle-hole-symmetric case there is always a quantum phase transition between Kondo and RKKY phases, even in the resonant regime (i.e. when impurity-host coupling exceeds Coulomb interaction strength). We show, 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 of the Kondo breakdown at a finite Kondo temperature, $T_K = T_K^0/e$, with $T_K$ meaning 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 Moire material for direct test of our predictions.

Related papers

Kondo screening and indirect magnetic exchange through a conventional superconductor studied by the density matrix renormalization group

Cassian Plorin, Michael Potthoff

The competition between the Kondo screening and indirect magnetic exchange in systems with two magnetic impurities coupled to a conventional $s$-wave superconductor gives rise to a nontrivial ground-state phase diagram. Here, we utilize the density matrix renormalization group (DMRG) method and exploit the non-Abelian spin-SU(2) symmetry to study the phase diagram for two quantum spin-$\frac{1}{2}$ impurities locally exchange coupled to large one-dimensional chains. The nonlocal interimpurity exchange is treated as an emergent Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling. We find qualitatively different phase diagrams for impurity spins coupled to sites with odd or even distances $d$ on the chain and a partial Kondo-screened spin-doublet phase that extends over the whole range of local exchange couplings $J$ in the limit of weak superconducting pairing strength $\mathrm{\ensuremath{\Delta}}$. Our numerical DMRG studies are complemented by exact diagonalization of small (quantum-box) systems and by perturbative-in-$J$ computations of the $d$- and $\mathrm{\ensuremath{\Delta}}$-dependent RKKY interaction. It is thereby demonstrated that the specific system geometry is essential for our understanding of magnetic impurity interactions in superconducting hosts, and thus for insights into the control of quantum-state properties in nanoparticle systems and topological superconductivity.

arXiv:2406.05029•Physical review. B./Physical review. B•Aug 9, 2024•1 citation

Multiple-impurity Anderson model for quantum dots coupled in parallel

Rok Žitko, J. Bonča

The system of several $(N)$ quantum dots coupled in parallel to the same single-mode conduction channel can be modeled as a single-channel $N$-impurity Anderson model. Using the generalized Schrieffer-Wolff transformation we show that near the particle-hole symmetric point, the effective Hamiltonian in the local moment regime is the $N$-impurity $S=1∕2$ Kondo model. The conduction-band-mediated RKKY exchange interaction between the dots is ferromagnetic and at intermediate temperatures locks the moments into a maximal spin $S=N∕2$ ground state. We provide an analytical estimate for the RKKY interaction. At low temperatures the spin is partially screened by the conduction electrons to $N∕2\ensuremath{-}1∕2$ due to the Kondo effect. By comparing accurate numerical renormalization group results for magnetic susceptibility of the $N$-impurity Anderson model to the exact Bethe ansatz results of a $S=N∕2$ SU(2) Kondo system we show that at low-temperature the quantum dots can be described by the effective $S=N∕2$ Kondo model. Moreover, the Kondo temperature is independent of the number of impurities $N$. We demonstrate the robustness of the spin $N∕2$ ground state as well as of the associated $S=N∕2$ Kondo effect by studying the stability of the system with respect to various experimentally relevant perturbations. We finally explore various quantum phase transitions driven by these perturbations.

arXiv:cond-mat/0604336•Physical Review B•Jul 17, 2006•118 citations

Two-channel multi-impurity Kondo model: RKKY-induced criticality

Alexei M. Tsvelik, Gabriel Kotliar

We show that the Ruderman-Kittel-Kasua-Yoisida interaction between overscreened spins in two-channel Kondo impurity systems is a relevant perturbation when the number of impurities N is greater than 3 driving the system to a new quantum critical point with anomalous dimensions $\frac{1}{(N+1)}$ for the spin operator and the Sommerfeld coefficient of the specific heat scales as $\ensuremath{\gamma}\ensuremath{\sim}{T}^{\ensuremath{-}\frac{3}{N+1}}$. The critical point universal properties are relevant to many strong correlation problems, such as impurity placed in a Majorana metal and the multichannel Kondo lattice model of heavy fermion materials. We discuss relevance of our results for cluster DMFT studies of quantum criticality.

Physical review. B./Physical review. B•Aug 7, 2025•0 citations

Two Kondo impurity problem: Relevance to heavy fermions

Barbara Jones, Chandra Mohan Varma

We have used Wilson's numerical renormalization group method to study the problem of two spin-one-half magnetic moments in a Fermi gas. Even when the initial effective RKKY coupling K0 between the impurities is much smaller than the Kondo energy TK, the system scales to strong coupling and the asymptotic behavior is that of a correlated Kondo effect.

Journal of Magnetism and Magnetic Materials•Jan 1, 1987•17 citations

Phase boundaries of power-law Anderson and Kondo models: A poor man's scaling study

Mengxing Cheng, Tathagata Chowdhury, Aaron S. Mohammed +1 more

We use the poor man's scaling approach to study the phase boundaries of a pair of quantum impurity models featuring a power-law density of states $\ensuremath{\rho}(\ensuremath{\varepsilon})\ensuremath{\propto}{|\ensuremath{\varepsilon}|}^{r}$, either vanishing (for $r>0)$ or diverging (for $r<0)$ at the Fermi energy $\ensuremath{\varepsilon}=0$, that gives rise to quantum phase transitions between local-moment and Kondo-screened phases. For the Anderson model with a pseudogap (i.e., $r>0)$, we find the phase boundary for (a) $0<r<1/2$, a range over which the model exhibits interacting quantum critical points both at and away from particle-hole (p-h) symmetry, and (b) $r>1$, where the phases are separated by first-order quantum phase transitions that are accessible only for broken p-h symmetry. For the p-h-symmetric Kondo model with easy-axis or easy-plane anisotropy of the impurity-band spin exchange, the phase boundary and scaling trajectories are obtained for both $r>0$ and $r<0$. Throughout the regime of weak-to-moderate impurity-band coupling in which poor man's scaling is expected to be valid, the approach predicts phase boundaries in excellent qualitative and good quantitative agreement with the nonperturbative numerical renormalization group, while also establishing the functional relations between model parameters along these boundaries.

arXiv:1702.06515•Physical review. B./Physical review. B•Jul 5, 2017•10 citations

Quantum phase transitions in models of coupled magnetic impurities

Matthias Vojta C, R. Bulla, Walter Hofstetter

We discuss models of interacting magnetic impurities coupled to a metallic host, which show one or more boundary quantum phase transitions where the ground-state spin changes as a function of the interimpurity couplings. The simplest example is realized by two spin-$\frac{1}{2}$ Kondo impurities coupled to a single orbital of the host. We investigate the phase diagram and crossover behavior of this model and present numerical renormalization-group results together with general arguments showing that the singlet-doublet quantum phase transition is either of first order or of the Kosterlitz-Thouless type, depending on the symmetry of the Kondo couplings. Thus we find an exponentially small energy scale within the Kondo regime, and a two-stage Kondo effect, in a single-channel situation. Connections to other models and possible applications are discussed.

arXiv:cond-mat/0106458•Physical review. B, Condensed matter•Apr 1, 2002•122 citations
Since Jones-Varma seminal works the indirect RKKY interaction is usually modeled as independently tunable Heisenberg spin exchange. For such a model in particle-hole-symmetric case there is always a quantum phase transition between Kondo and RKKY phases, even in the resonant regime (i.e. when impurity-host coupling exceeds Coulomb interaction strength). We show, 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 of the Kondo breakdown at a finite Kondo temperature, TK=TK0/eT_K = T_K^0/eTK​=TK0​/e, with TKT_KTK​ meaning 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 Moire material for direct test of our predictions.
The competition between the Kondo screening and indirect magnetic exchange in systems with two magnetic impurities coupled to a conventional sss-wave superconductor gives rise to a nontrivial ground-state phase diagram. Here, we utilize the density matrix renormalization group (DMRG) method and exploit the non-Abelian spin-SU(2) symmetry to study the phase diagram for two quantum spin-12\frac{1}{2}21​ impurities locally exchange coupled to large one-dimensional chains. The nonlocal interimpurity exchange is treated as an emergent Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling. We find qualitatively different phase diagrams for impurity spins coupled to sites with odd or even distances ddd on the chain and a partial Kondo-screened spin-doublet phase that extends over the whole range of local exchange couplings JJJ in the limit of weak superconducting pairing strength $\mathrm{\ensuremath{\Delta}}$. Our numerical DMRG studies are complemented by exact diagonalization of small (quantum-box) systems and by perturbative-in-JJJ computations of the ddd- and $\mathrm{\ensuremath{\Delta}}$-dependent RKKY interaction. It is thereby demonstrated that the specific system geometry is essential for our understanding of magnetic impurity interactions in superconducting hosts, and thus for insights into the control of quantum-state properties in nanoparticle systems and topological superconductivity.
The system of several (N)(N)(N) quantum dots coupled in parallel to the same single-mode conduction channel can be modeled as a single-channel NNN-impurity Anderson model. Using the generalized Schrieffer-Wolff transformation we show that near the particle-hole symmetric point, the effective Hamiltonian in the local moment regime is the NNN-impurity S=1∕2S=1∕2S=1∕2 Kondo model. The conduction-band-mediated RKKY exchange interaction between the dots is ferromagnetic and at intermediate temperatures locks the moments into a maximal spin S=N∕2S=N∕2S=N∕2 ground state. We provide an analytical estimate for the RKKY interaction. At low temperatures the spin is partially screened by the conduction electrons to $N∕2\ensuremath{-}1∕2$ due to the Kondo effect. By comparing accurate numerical renormalization group results for magnetic susceptibility of the NNN-impurity Anderson model to the exact Bethe ansatz results of a S=N∕2S=N∕2S=N∕2 SU(2) Kondo system we show that at low-temperature the quantum dots can be described by the effective S=N∕2S=N∕2S=N∕2 Kondo model. Moreover, the Kondo temperature is independent of the number of impurities NNN. We demonstrate the robustness of the spin N∕2N∕2N∕2 ground state as well as of the associated S=N∕2S=N∕2S=N∕2 Kondo effect by studying the stability of the system with respect to various experimentally relevant perturbations. We finally explore various quantum phase transitions driven by these perturbations.
We show that the Ruderman-Kittel-Kasua-Yoisida interaction between overscreened spins in two-channel Kondo impurity systems is a relevant perturbation when the number of impurities N is greater than 3 driving the system to a new quantum critical point with anomalous dimensions 1(N+1)\frac{1}{(N+1)}(N+1)1​ for the spin operator and the Sommerfeld coefficient of the specific heat scales as $\ensuremath{\gamma}\ensuremath{\sim}{T}^{\ensuremath{-}\frac{3}{N+1}}$. The critical point universal properties are relevant to many strong correlation problems, such as impurity placed in a Majorana metal and the multichannel Kondo lattice model of heavy fermion materials. We discuss relevance of our results for cluster DMFT studies of quantum criticality.
We use the poor man's scaling approach to study the phase boundaries of a pair of quantum impurity models featuring a power-law density of states $\ensuremath{\rho}(\ensuremath{\varepsilon})\ensuremath{\propto}{|\ensuremath{\varepsilon}|}^{r}$, either vanishing (for r>0)r>0)r>0) or diverging (for r<0)r<0)r<0) at the Fermi energy $\ensuremath{\varepsilon}=0$, that gives rise to quantum phase transitions between local-moment and Kondo-screened phases. For the Anderson model with a pseudogap (i.e., r>0)r>0)r>0), we find the phase boundary for (a) 0<r<1/20<r<1/20<r<1/2, a range over which the model exhibits interacting quantum critical points both at and away from particle-hole (p-h) symmetry, and (b) r>1r>1r>1, where the phases are separated by first-order quantum phase transitions that are accessible only for broken p-h symmetry. For the p-h-symmetric Kondo model with easy-axis or easy-plane anisotropy of the impurity-band spin exchange, the phase boundary and scaling trajectories are obtained for both r>0r>0r>0 and r<0r<0r<0. Throughout the regime of weak-to-moderate impurity-band coupling in which poor man's scaling is expected to be valid, the approach predicts phase boundaries in excellent qualitative and good quantitative agreement with the nonperturbative numerical renormalization group, while also establishing the functional relations between model parameters along these boundaries.
We discuss models of interacting magnetic impurities coupled to a metallic host, which show one or more boundary quantum phase transitions where the ground-state spin changes as a function of the interimpurity couplings. The simplest example is realized by two spin-12\frac{1}{2}21​ Kondo impurities coupled to a single orbital of the host. We investigate the phase diagram and crossover behavior of this model and present numerical renormalization-group results together with general arguments showing that the singlet-doublet quantum phase transition is either of first order or of the Kosterlitz-Thouless type, depending on the symmetry of the Kondo couplings. Thus we find an exponentially small energy scale within the Kondo regime, and a two-stage Kondo effect, in a single-channel situation. Connections to other models and possible applications are discussed.