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.
The existence of a length scale ${\ensuremath{\xi}}_{K}\ensuremath{\sim}1/{T}_{K}$ (with
TK the Kondo temperature) has long been predicted in quantum impurity systems. At low temperatures $T\ensuremath{\ll}{T}_{K}$, the standard interpretation is that a spin-
21 impurity is screened by a surrounding ``Kondo cloud'' of spatial extent ${\ensuremath{\xi}}_{K}$. We argue that renormalization group (RG) flow between any two fixed points (FPs) results in a characteristic length scale, observed in real space as a crossover between physical behavior typical of each FP. In the simplest example of the Anderson impurity model, three FPs arise, and we show that ``free orbital,'' ``local moment,'' and ``strong coupling'' regions of space can be identified at zero temperature. These regions are separated by two crossover length scales ${\ensuremath{\xi}}_{\text{LM}}$ and ${\ensuremath{\xi}}_{K}$, with the latter diverging as the Kondo effect is destroyed on increasing temperature through
TK. One implication is that moment formation occurs inside the ``Kondo cloud'', while the screening process itself occurs on flowing to the strong coupling FP at distances $\ensuremath{\sim}{\ensuremath{\xi}}_{K}$. Generic aspects of the real-space physics are exemplified by the two-channel Kondo model, where ${\ensuremath{\xi}}_{K}$ now separates local moment and overscreening clouds.
Quantum impurity models are the prototypical examples of quantum many-body dynamics which manifests in their spectral and transport properties. Single channel Anderson(and Kondo model) leads to the Fermi liquid ground state in the strong coupling regime which corresponds to a stable infrared fixed point at which the quantum impurity gets completely screened by the conduction electrons. Quantum impurity models with non-trivial density of states exhibit quantum phase transition and this quantum criticality lies in the universality class of local quantum critical systems. In this paper, we report first study of the flow equation renormalization of gapped Kondo model which has gapped density of states, the gap being at the Fermi level. Flow equation renormalization group method has proved to be one of the very robust renormalization methods to study Kondo physics both in equilibrium as well as in non-equilibrium. Here we have shown that this method can also be employed to study local quantum criticality. We have calculated the flow equations for the Kondo coupling and solved them for various values of the gap parameter and we find that there is suppression of Kondo divergence as gap is increased which signifies that as gap is increased, renormalization flow goes away from the strong coupling fixed point. We have also calculated the spin susceptibility and we find that as gap is increased, susceptibility goes over to the Curie behaviour and hence confirming the renormalization flow towards the local moment fixed point.
The Kondo problem, which describes the interaction of a spin
s magnetic impurity with a free Fermi gas, is a classic example of strongly coupled physics. Historically, the problem has been solved by Wilson's numerical renormalization group and later by Bethe ansatz. In this paper, we present an alternate analytic solution of the Kondo problem that combines an expansion in
1/s with the renormalization group. We study both the case of an impurity interacting with a single channel
K=1 of fermions in the
s→∞ limit and the case with
K channels in the double-scaling limit
K→∞,
s→∞,
K/s fixed. Our approach allows us to describe analytically intermediate scales of the Kondo problem at large
s and compute thermodynamic observables such as the impurity entropy and susceptibility. We find these observables to agree with the Bethe ansatz results. We also compute the impurity spectral function, finite temperature resistivity and the Kondo screening cloud profile, properties that are not easily accessible from Bethe ansatz. Notably, in the regime
K>2s we access the "non-Fermi-liquid" overscreened fixed point of the multichannel Kondo problem.
We disentangle all the individual degrees of freedom in the quantum impurity problem to deconstruct the Kondo singlet, both in real and energy space, by studying the contribution of each individual free electron eigenstate. This is a problem of two spins coupled to a bath, where the bath is formed by the remaining conduction electrons. Being a mixed state, we resort to the ``concurrence'' to quantify entanglement. We identify ``projected natural orbitals'' that allow us to individualize a single-particle electronic wave function that is responsible of more than
90 of the impurity screening. In the weak coupling regime, the impurity is entangled to an electron at the Fermi level, while in the strong coupling regime, the impurity counterintuitively entangles mostly with the high energy electrons and disentangles completely from the low-energy states carving a ``hole'' around the Fermi level. This enables one to use concurrence as a pseudo order parameter to compute the characteristic ``size'' of the Kondo cloud, beyond which electrons are weakly correlated to the impurity and are dominated by the physics of the boundary.
The two-channel Kondo (2CK) effect with its exotic ground state properties has remained difficult to realize in physical systems. At low energies, a quantum impurity with orbital degree of freedom, like a proton bound in an interstitial lattice space, comprises a 3-level system with a unique ground state and (at least) doubly degenerate rotational excitations with excitation energy
Delta0. When immersed in a metal, electronic angular momentum scattering induces transitions between any two of these levels (couplings
J), while the electron spin is conserved. We show by extensive numerical renormalization group (NRG) calculations that without fine-tuning of parameters this system exhibits a 2CK fixed point, due to Kondo correlations in the excited-state doublet whose degeneracy is stabilized by the host lattice parity, while the channel symmetry (electron spin) is guaranteed by time reversal symmetry. We find a pronounced plateau in the entropy at
S(TK<T<Delta0)=kBln2 between the high-
T value,
S(TggDelta0)=kBln3, and the 2CK ground state value,
S(0)=kBlnsqrt2. This indicates a downward renormalization of the doublet below the non-interacting ground state, thus realizing the 2CK fixed point, in agreement with earlier conjectures. We mapped out the phase diagram of the model in the
J−Delta0 plane. The Kondo temperature
TK shows non-monotonic
J-dependence, characteristic for 2CK systems.newline indent Beside the two-channel Kondo effect of the model, we also study the single-channel version, which is realized by applying a strong magnetic field to the conduction band electrons so that their degeneracy is lifted and consequently having only one kind of electrons scattering off the impurity. This single-channel case is easier to analyze since the Hilbert space is not as large as that of the 2CK. We equally find a downward renormalization of the excited state energy by the Kondo correlations in the SU(2) doublet. In a wide range of parameter values this stabilizes the single-channel Kondo fixed point and a phase diagram is also mapped out for the model. In the single-channel version a plateau is found in the entropy at
S(TK<T<Delta0)=kBln2 between high-
T value,
S(TggDelta0)=kBln3, and the single-channel Kondo ground state value,
S(0)=kBln1.