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

Physical review. B./Physical review. B

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

Alexei M. Tsvelik, Gabriel Kotliar

Published Aug 7, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1103/ty27-vl8n

Related papers

Two Channel Multi impurity Kondo model

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 $γ\sim T^{- \frac{3}{N+1}}$. 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.

arXiv:2502.07506•Feb 11, 2025•0 citations

Entanglement Probe of Two-Impurity Kondo Physics in a Spin Chain

Abolfazl Bayat, Sougato Bose, Pasquale Sodano +1 more

We propose that real-space properties of the two-impurity Kondo model can be obtained from an effective spin model where two single-impurity Kondo spin chains are joined via an Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between the two impurity spins. We then use a density matrix renormalization group approach, valid in all ranges of parameters, to study its features using two complementary quantum-entanglement measures, the negativity and the von Neumann entropy. This nonperturbative approach enables us to uncover the precise dependence of the spatial extent ξ(K) of the Kondo screening cloud with the Kondo and RKKY couplings. Our results reveal an exponential suppression of the Kondo temperature T(K)~1/ξ(K) with the size of the effective impurity spin in the limit of large ferromagnetic RKKY coupling, a striking display of "Kondo resonance narrowing" in the two-impurity Kondo model. We also show how the antiferromagnetic RKKY interaction produces an effective decoupling of the impurities from the bulk already for intermediate strengths of this interaction, and, furthermore, exhibit how the non-Fermi liquid quantum critical point is signaled in the quantum entanglement between various parts of the system.

arXiv:1201.6668•Physical Review Letters•Aug 7, 2012•99 citations

Overscreened multichannel SU(N) Kondo model: Large- N solution and conformal field theory

Olivier Parcollet, Antoine Georges, Gabriel Kotliar +1 more

The multichannel Kondo model with $\mathrm{SU}(N)$ spin symmetry and $\mathrm{SU}(K)$ channel symmetry is considered. The impurity spin is chosen to transform as an antisymmetric representation of $\mathrm{SU}(N),$ corresponding to a fixed number of Abrikosov fermions ${\ensuremath{\sum}}_{\ensuremath{\alpha}}{f}_{\ensuremath{\alpha}}^{\ifmmode\dagger\else\textdagger\fi{}}{f}_{\ensuremath{\alpha}}=Q.$ For more than one channel $(K>1),$ and all values of $N$ and $Q,$ the model displays non-Fermi behavior associated with the overscreening of the impurity spin. Universal low-temperature thermodynamic and transport properties of this non-Fermi-liquid state are computed using conformal field theory methods. A large-$N$ limit of the model is then considered, in which $K/N\ensuremath{\equiv}\ensuremath{\gamma}$ and $Q/N\ensuremath{\equiv}{q}_{0}$ are held fixed. Spectral densities satisfy coupled integral equations in this limit, corresponding to a (time-dependent) saddle point. A low-frequency, low-temperature analysis of these equations reveals universal scaling properties in the variable $\ensuremath{\omega}/T,$ in agreement with conformal invariance. The universal scaling form is obtained analytically and used to compute the low-temperature universal properties of the model in the large-$N$ limit, such as the $T=0$ residual entropy and residual resistivity, and the critical exponents associated with the specific heat and susceptibility. The connections with the ``noncrossing approximation'' and the previous work of Cox and Ruckenstein are discussed.

arXiv:cond-mat/9711192•Physical review. B, Condensed matter•Aug 15, 1998•180 citations

Relevance of anisotropy in the multichannel Kondo effect: Comparison of conformal field theory and numerical renormalization-group results

Ian K. Affleck, Andreas W. W. Ludwig, H.-B. Pang +1 more

The multichannel Kondo model exhibits non-Fermi-liquid behavior in the overscreened case, when the number of channels, k, is greater than twice the size of the impurity spin, s. We show that, for overscreening, exchange anisotropy is irrelevant at the low-temperature fixed point for s=1/2 or s=k/2-1/2, but relevant for all other values of s. However, an external field or channel asymmetry is relevant, producing crossover to Fermi-liquid fixed points with different phase shifts for each spin in the first case and for each channel in the second. These results are elucidated by explicit comparison of analytic finite-size spectra derived from conformal field theory with those obtained from numerical renormalization-group calculations. The relevance of the results (for k=2 and s=1/2) to the quadrupolar Kondo Hamiltonian (which has been proposed as a model for many uranium-based heavy-fermion materials) will be briefly discussed. Also, a larger, a/Isymplectic symmetry Sp(2k) is shown to be present in the k-channel Kondo model.

Physical review. B, Condensed matter•Apr 1, 1992•204 citations

RKKY interaction and two Kondo impurities: the complete phase diagram

Krzysztof P. Wójcik, Johann Kroha

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.

•Jun 14, 2021•0 citations

Quantum entanglement in the two-impurity Kondo model

Sam Young Cho, Ross H. McKenzie

In order to quantify quantum entanglement in two-impurity Kondo systems, we calculate the concurrence, negativity, and von Neumann entropy. The entanglement of the two Kondo impurities is shown to be determined by two competing many-body effects, namely the Kondo effect and the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, $I$. Due to the spin-rotational invariance of the ground state, the concurrence and negativity are uniquely determined by the spin-spin correlation between the impurities. It is found that there exists a critical minimum value of the antiferromagnetic correlation between the impurity spins which is necessary for entanglement of the two impurity spins. The critical value is discussed in relation with the unstable fixed point in the two-impurity Kondo problem. Specifically, at the fixed point there is no entanglement between the impurity spins. Entanglement will only be created [and quantum information processing (QIP) will only be possible] if the RKKY interaction exchange energy, $I$, is at least several times larger than the Kondo temperature, ${T}_{K}$. Quantitative criteria for QIP are given in terms of the impurity spin-spin correlation.

arXiv:quant-ph/0509207•Physical Review A•Jan 19, 2006•63 citations
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 \ensuremathγ\ensuremath∼T\ensuremath−3N+1\ensuremath{\gamma}\ensuremath{\sim}{T}^{\ensuremath{-}\frac{3}{N+1}}\ensuremathγ\ensuremath∼T\ensuremath−N+13​. 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 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 γ∼T−3N+1γ\sim T^{- \frac{3}{N+1}}γ∼T−N+13​. 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 propose that real-space properties of the two-impurity Kondo model can be obtained from an effective spin model where two single-impurity Kondo spin chains are joined via an Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between the two impurity spins. We then use a density matrix renormalization group approach, valid in all ranges of parameters, to study its features using two complementary quantum-entanglement measures, the negativity and the von Neumann entropy. This nonperturbative approach enables us to uncover the precise dependence of the spatial extent ξ(K) of the Kondo screening cloud with the Kondo and RKKY couplings. Our results reveal an exponential suppression of the Kondo temperature T(K)~1/ξ(K) with the size of the effective impurity spin in the limit of large ferromagnetic RKKY coupling, a striking display of "Kondo resonance narrowing" in the two-impurity Kondo model. We also show how the antiferromagnetic RKKY interaction produces an effective decoupling of the impurities from the bulk already for intermediate strengths of this interaction, and, furthermore, exhibit how the non-Fermi liquid quantum critical point is signaled in the quantum entanglement between various parts of the system.
The multichannel Kondo model with SU(N)\mathrm{SU}(N)SU(N) spin symmetry and SU(K)\mathrm{SU}(K)SU(K) channel symmetry is considered. The impurity spin is chosen to transform as an antisymmetric representation of SU(N),\mathrm{SU}(N),SU(N), corresponding to a fixed number of Abrikosov fermions ${\ensuremath{\sum}}_{\ensuremath{\alpha}}{f}_{\ensuremath{\alpha}}^{\ifmmode\dagger\else\textdagger\fi{}}{f}_{\ensuremath{\alpha}}=Q.$ For more than one channel (K>1),(K>1),(K>1), and all values of NNN and Q,Q,Q, the model displays non-Fermi behavior associated with the overscreening of the impurity spin. Universal low-temperature thermodynamic and transport properties of this non-Fermi-liquid state are computed using conformal field theory methods. A large-NNN limit of the model is then considered, in which $K/N\ensuremath{\equiv}\ensuremath{\gamma}$ and $Q/N\ensuremath{\equiv}{q}_{0}$ are held fixed. Spectral densities satisfy coupled integral equations in this limit, corresponding to a (time-dependent) saddle point. A low-frequency, low-temperature analysis of these equations reveals universal scaling properties in the variable $\ensuremath{\omega}/T,$ in agreement with conformal invariance. The universal scaling form is obtained analytically and used to compute the low-temperature universal properties of the model in the large-NNN limit, such as the T=0T=0T=0 residual entropy and residual resistivity, and the critical exponents associated with the specific heat and susceptibility. The connections with the ``noncrossing approximation'' and the previous work of Cox and Ruckenstein are discussed.
The multichannel Kondo model exhibits non-Fermi-liquid behavior in the overscreened case, when the number of channels, k, is greater than twice the size of the impurity spin, s. We show that, for overscreening, exchange anisotropy is irrelevant at the low-temperature fixed point for s=1/2 or s=k/2-1/2, but relevant for all other values of s. However, an external field or channel asymmetry is relevant, producing crossover to Fermi-liquid fixed points with different phase shifts for each spin in the first case and for each channel in the second. These results are elucidated by explicit comparison of analytic finite-size spectra derived from conformal field theory with those obtained from numerical renormalization-group calculations. The relevance of the results (for k=2 and s=1/2) to the quadrupolar Kondo Hamiltonian (which has been proposed as a model for many uranium-based heavy-fermion materials) will be briefly discussed. Also, a larger, a/Isymplectic symmetry Sp(2k) is shown to be present in the k-channel Kondo model.
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.
In order to quantify quantum entanglement in two-impurity Kondo systems, we calculate the concurrence, negativity, and von Neumann entropy. The entanglement of the two Kondo impurities is shown to be determined by two competing many-body effects, namely the Kondo effect and the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, III. Due to the spin-rotational invariance of the ground state, the concurrence and negativity are uniquely determined by the spin-spin correlation between the impurities. It is found that there exists a critical minimum value of the antiferromagnetic correlation between the impurity spins which is necessary for entanglement of the two impurity spins. The critical value is discussed in relation with the unstable fixed point in the two-impurity Kondo problem. Specifically, at the fixed point there is no entanglement between the impurity spins. Entanglement will only be created [and quantum information processing (QIP) will only be possible] if the RKKY interaction exchange energy, III, is at least several times larger than the Kondo temperature, TK{T}_{K}TK​. Quantitative criteria for QIP are given in terms of the impurity spin-spin correlation.