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arXiv:2608.14072

Quantum Geometric Kondo Cloud

Grant Z. X. Yang, K. T. Law

Published Aug 14, 2026• 0 citations
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Abstract

A magnetic impurity embedded in a metal is collectively screened by Fermi-surface quasiparticles into a many-body spin-singlet ground state, forming a Kondo cloud of size $ξ_{\rm K}\sim\hbar v_F/(k_B T_{\rm K})$. This kinematic picture collapses in flat bands, where $v_F=0$ and the hierarchy of dispersive energy shells is absent. Here we show that the missing organizing principle is quantum geometry. A magnetic impurity coupled to an isolated flat band selects a single active bath mode: a coherent superposition of flat-band Bloch states weighted by the hybridization factor $v(\mathbf{k})$, while all orthogonal flat-band modes remain dark. The resulting flat-band Kondo problem is a quantum geometric molecule, with an algebraic Kondo scale set by the total projected hybridization strength rather than a logarithmic-renormalization scale. In real space, the impurity-bath spin correlation defines a quantum geometric Kondo cloud. Its cloud-size tensor admits a gauge-invariant decomposition into a hybridization-weighted quantum metric, a dressed Berry-connection covariance, and a positive hybridization-gradient term, yielding the lower bound $ξ_{\rm K}^2\geq \sum_{\mathbf{k}}ρ(\mathbf{k}){\rm Tr}\,g(\mathbf{k})$, where $ρ(\mathbf{k})=|v(\mathbf{k})|^2/\sum_{\mathbf{k}}|v(\mathbf{k})|^2$. Our result reveals that, in flat bands, Kondo screening is governed by the quantum geometry and interference structure of the impurity-selected Bloch wave packet, rather than Fermi-surface kinematics.

DOI
10.48550/arxiv.2608.14072

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A magnetic impurity embedded in a metal is collectively screened by Fermi-surface quasiparticles into a many-body spin-singlet ground state, forming a Kondo cloud of size ξK∼ℏvF/(kBTK)ξ_{\rm K}\sim\hbar v_F/(k_B T_{\rm K})ξK​∼ℏvF​/(kB​TK​). This kinematic picture collapses in flat bands, where vF=0v_F=0vF​=0 and the hierarchy of dispersive energy shells is absent. Here we show that the missing organizing principle is quantum geometry. A magnetic impurity coupled to an isolated flat band selects a single active bath mode: a coherent superposition of flat-band Bloch states weighted by the hybridization factor v(k)v(\mathbf{k})v(k), while all orthogonal flat-band modes remain dark. The resulting flat-band Kondo problem is a quantum geometric molecule, with an algebraic Kondo scale set by the total projected hybridization strength rather than a logarithmic-renormalization scale. In real space, the impurity-bath spin correlation defines a quantum geometric Kondo cloud. Its cloud-size tensor admits a gauge-invariant decomposition into a hybridization-weighted quantum metric, a dressed Berry-connection covariance, and a positive hybridization-gradient term, yielding the lower bound ξK2≥∑kρ(k)Tr g(k)ξ_{\rm K}^2\geq \sum_{\mathbf{k}}ρ(\mathbf{k}){\rm Tr}\,g(\mathbf{k})ξK2​≥∑k​ρ(k)Trg(k), where ρ(k)=∣v(k)∣2/∑k∣v(k)∣2ρ(\mathbf{k})=|v(\mathbf{k})|^2/\sum_{\mathbf{k}}|v(\mathbf{k})|^2ρ(k)=∣v(k)∣2/∑k​∣v(k)∣2. Our result reveals that, in flat bands, Kondo screening is governed by the quantum geometry and interference structure of the impurity-selected Bloch wave packet, rather than Fermi-surface kinematics.
We analyze the single-channel Kondo model using the recently developed unitary renormalization group (URG) method, and obtain a comprehensive understanding of the Kondo screening cloud. The fixed-point low-energy Hamiltonian enables the computation of a plethora of thermodynamic quantities (specific heat, susceptibility, Wilson ratio, etc.) as well as spectral functions, all of which are found to be in excellent agreement with known results. By integrating out the impurity, we obtain an effective Hamiltonian for the excitations of the electrons comprising the Kondo cloud. This is found to contain both k-space number-diagonal (Fermi liquid) as well off-diagonal four-fermion scattering terms. Our conclusions are reinforced by a URG study of the two-particle entanglement and many-body correlations among members of the Kondo cloud and impurity. The entanglement between the impurity and a cloud electron, as well as between any two cloud electrons, is found to increase under flow towards the singlet ground state at the strong-coupling fixed point. Both the number-diagonal and off-diagonal correlations within the conduction cloud are also found to increase as the impurity is screened under the flow, and the latter are found to be responsible for the macroscopic entanglement of the Kondo-singlet ground state. The unitary RG flow enables an analytic computation of the phase shifts suffered by the conduction electrons at the strong-coupling fixed point. This reveals an orthogonality catastrophe between the local moment and strong-coupling ground states, and is related to a change in the Luttinger volume of the conduction bath. Our results offer fresh insight on the nature of the emergent many-particle entanglement within the Kondo cloud, and pave the way for further investigations in more exotic contexts such as the fixed point of the over-screened multi-channel Kondo problem.
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We analyze the single-channel Kondo model using the recently developed unitary renormalization-group (URG) method and obtain a comprehensive understanding of the Kondo screening cloud. The fixed-point low-energy Hamiltonian enables the computation of a plethora of thermodynamic quantities (specific heat, susceptibility, Wilson ratio, etc.) as well as spectral functions, all of which are found to be in good agreement with known results. By integrating out the impurity, we obtain an effective Hamiltonian for the excitations of the electrons comprising the Kondo cloud. This is found to contain both kkk-space number diagonal (Fermi liquid) and off-diagonal four-fermion scattering terms. Our conclusions are reinforced by a URG study of the two-particle entanglement and many-body correlations among members of the Kondo cloud and impurity. The entanglement between the impurity and a cloud electron, as well as between any two cloud electrons, is found to increase under flow towards the singlet ground state at the strong-coupling fixed point. Both the number diagonal and off-diagonal correlations within the conduction cloud are also found to increase as the impurity is screened under the flow, and the latter are found to be responsible for the macroscopic entanglement of the Kondo-singlet ground state. The URG flow enables an analytic computation of the phase shifts suffered by the conduction electrons at the strong-coupling fixed point. This reveals an orthogonality catastrophe between the local-moment and strong-coupling ground states and is related to a change in the Luttinger volume of the conduction bath under the crossover to strong coupling. Our results offer fresh insight into the nature of the emergent many-particle entanglement within the Kondo cloud and pave the way for further investigations in more exotic contexts such as the fixed point of the overscreened multichannel Kondo problem.
Abstract The screening of an impurity spin in the Kondo model occurs over a characteristic length scale ξ K , that defines the size of the Kondo screening cloud or ‘mist’. The presence of such a length scale in experimental and numerical results is rather subtle. A consistent way to show the presence of the screening cloud is to demonstrate scaling in the spatial correlations in terms of the single variable, r / ξ K instead of the correlations depending on r and ξ K separately. Here we study the paradigmatic one channel Kondo model using a spin chain representation, with an impurity spin at one end of the chain coupled with a strength J K ′ . Using Fermi liquid theory combined with numerical results, we obtain new high precision estimates of the non-universal terms in the entanglement entropy which leads to a verification of the expected non-integer ground-state degeneracy, g. This then allows us to study the impurity contribution to the entanglement in detail. If the impurity coupling J K ′ is varied, a precise determination of ξ K can then be obtained. The length scale, ξ K , is then shown to characterize the scaling of both the uniform and alternating part of a measure of the magnetization of part of an odd length chain.
The Kondo effect of a single magnetic impurity embedded in the Lieb lattice is studied by the numerical renormalization group. When the band flatness is present in the local density of states at the impurity site, it quenches the participation of all dispersive electrons in the Kondo singlet formation and reduces the many-body Kondo problem to a two-electron molecular Kondo problem. A quantum entanglement of two spins, which is the two-electron molecular analog of the many-body Kondo singlet, is stable at low temperature, and the impurity contributions to thermodynamical and dynamical quantities are qualitatively different from that obtained in the many-body Kondo effect. The conditions for existence of the molecular Kondo effect in narrow band systems are also presented.
One of the most striking aspects of a flat band is not merely its lack of dispersion, but that it reduces the single-particle kinetic energy to a very small scale, allowing interaction effects to become the dominant energy scale in the low-energy physics. This picture has underpinned concepts such as flat-band ferromagnetism, flat-band superconductivity, and localized magnons. A more fundamental question, however, remains. Flat bands belong to the language of single-particle or linear-spin-wave physics; whether they survive as protected sharp excitations in a genuine many-body quantum magnet, or merely provide a skeleton of spectral weight left by the noninteracting picture, remains unclear. We study the spin-1/2 Heisenberg antiferromagnet on the edge-centered honeycomb lattice to examine this question. In contrast to flat bands in geometrically frustrated magnets such as kagome, this lattice has a 2:3 sublattice imbalance that renders the intersublattice coupling matrix rectangular, whose null space gives rise to a flat band at the single-particle level. The same imbalance stabilizes a ferrimagnetic ground state through the Lieb--Mattis theorem and leads to an edge-site-localized flat magnon in linear spin-wave theory. Combining linear spin-wave theory, sign-problem-free stochastic series expansion quantum Monte Carlo, and stochastic analytic continuation, we find that quantum fluctuations transform this harmonic flat mode into a broad continuum whose centroid is about 1.55J and nearly dispersionless along the high-symmetry path. Sublattice imbalance thus offers an unfrustrated route that unifies flat bands and ferrimagnetism, and suggests that flat-band geometry organizes many-body spectral weight rather than giving rise to a protected quasiparticle.