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
k-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.
The Kondo problem concerns a simple physical model of magnetic impurities in metals that has required the most sophisticated of mathematical techniques. To theorists, the model not only represents a first step towards understanding the far more complex problem of ferromagnetism but also appears tantalizingly close to exact solution by the techniques employed for renormalizable field theories. In 1975 Kenneth Wilson of Cornell University successfully used renormalization-group techniques to calculate numerically a key parameter in the Kondo problem. This past year Natan Andrei of New York University and independently P. B. Wiegmann of the Landau Institute in Moscow executed a second coup by exactly diagonalizing the Hamiltonian to allow a solution in closed form.
A path integral formulation of the SU(N) Kondo model is explored as a possible theoretical tool for studying generalizations of the model, such as the Kondo lattice, which are difficult to study by other means. Saddle point approximations to the path integral lead to an expansion in powers of 1/N. After developing the basic path integral formalism, the method is applied to three related problems. The first of these is an exactly soluble toy model which, like the Kondo model, shows smooth crossover between weakly mixed configurations at high energy scales and strongly mixed singlet configurations at low energy scales. Simple approximations are found for describing the high and low temperature limits of the model, and the failure of these approximations in the crossover region is interpreted in terms of differences between the high and low energy scales. In chapter 4 the low temperature approximation is used to study the Kondo crossover as a function of magnetic field at zero temperature. Smooth and plausible crossover is found provided the magnetic field is introduced in a way which preserves degeneracy proportional toN, indicating that the failure to describe temperature crossover is not a general limitation of the method. Finally, the same zero temperature equations are generalized in chapter 5 to a model of a magnetic impurity interacting with a gapless electronic band. Using a simple renormalization group argument this model is shown to have a transition as a function of coupling constant at zero temperature. This result is confirmed and extended using the zero temperature path integral equations, and certain special features of the magnetic field crossover are noted.
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 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.
The Kondo model describes a magnetic impurity in a metallic host. Here, the authors use variational approaches (Lanczos, Yosida, Gutzwiller) and quasi-exact numerical techniques (numerical renormalization group, density matrix renormalization group) to study the ground-state energy and magnetization, and compare these properties to Bethe ansatz results. The Gutzwiller wave function is an excellent trial state for strong coupling. For weak coupling, the numerical renormalization group gives the best description of the magnetic properties. A combination of Bethe ansatz and perturbation theory provides new benchmark results for the zero-field susceptibility.