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Journal of Physics C Solid State Physics

Universality and the renormalisability of rotationally invariant Kondo Hamiltonians

Dinah M. Cragg, P Lloyd

Published Aug 28, 1979• 21 citations
Publisher

Abstract

Wilson's numerical technique (1974, 1975) has been used to study the effects of including an extra potential term in the Kondo Hamiltonian and the effects of using and s-d exchange matrix which lacks particle-hole symmetry. It is found that these generalised Kondo Hamiltonians have universal properties. They can all be mapped into a canonical form for the Kondo Hamiltonian, one which possesses a particle-hole symmetry, provided that the operators in this canonical form now create and destroy one-electron basis states that have a phase-shift with respect to the original ones. The underlying physical reason for this universality is that the field theory belonging to the Kondo Hamiltonian is renormalisable.

DOI
10.1088/0022-3719/12/16/018

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Wilson's numerical technique (1974, 1975) has been used to study the effects of including an extra potential term in the Kondo Hamiltonian and the effects of using and s-d exchange matrix which lacks particle-hole symmetry. It is found that these generalised Kondo Hamiltonians have universal properties. They can all be mapped into a canonical form for the Kondo Hamiltonian, one which possesses a particle-hole symmetry, provided that the operators in this canonical form now create and destroy one-electron basis states that have a phase-shift with respect to the original ones. The underlying physical reason for this universality is that the field theory belonging to the Kondo Hamiltonian is renormalisable.
Whereas the standard expansion of the 'quantum field theory' belonging to the s-d exchange Kondo Hamiltonian for a spin impurity in a paramagnetic host metal describes the high-energy scattering, the alternative renormalised perturbation expansion describes the low-energy scattering. In the high-energy regime the scattering centre acts as if it was an anti-ferromagnetically coupled spin, but in the low-energy regime it acts as if it was a spin of a different magnitude that is ferromagnetically coupled. Correspondingly, it is shown that two different Hamiltonians can give rise to the same field theory. In solid state physics this equivalence is the generalisation of the Schrieffer-Wolff transformation (1960) between the Anderson (or Wolff) model and the spin- 1 / 2 Kondo Hamiltonian.
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.
We discuss the scale-free property of Wilson's numerical renormalization group (NRG) for the Kondo-impurity problem. The single-particle state of the effective Hamiltonian with a cutoff $\ensuremath{\Lambda}$ is described by the wave-packet basis having the scale-free property. The energy scale of the system can be controlled by the lattice translation of the wave-packet basis with no reference of rescaling of the lattice space. We also analyze the role of the Kondo interaction in the context of wave-packet basis and then discuss the scaling and renormalization of the Kondo coupling. In addition, we clarify the role of the edge state in the lowest energy scale of Wilson NRG.
Numerical renormalization-group and conformal-field-theory work indicate that the two-impurity Kondo Hamiltonian has a non-Fermi-liquid critical point separating the Kondo-screening phase from the interimpurity singlet phase when particle-hole (P-H) symmetry is maintained. We clarify the circumstances under which this critical point occurs, pointing out that there are two types of P-H symmetry. Only one of them guarantees the occurrence of the critical point. Much of the previous numerical work was done on models with the other type of P-H symmetry. We analyze this critical point using the boundary conformal-field-theory technique. The finite-size spectrum is presented in detail and compared with about 50 energy levels obtained using the numerical renormalization group. Various Green's functions, general renormalization group behavior, and a hidden SO(7) are analyzed.
Entanglement of spin and orbital Kondo effect is investigated on the basis of a Kondo-type exchange model with twofold orbital degeneracy. By using Wilson's numerical renormalization-group method, we examine dynamical and thermal properties respecting the difference in time-reversal property of multipole operators. In the presence of particle-hole symmetry, the model has a new non-Fermi-liquid fixed point with a fractional entropy. The spectral intensity of the quadrupole susceptibility diverges in the zero-frequency limit, while the dipole susceptibility shows a Fermi-liquid-like behavior. This is understood by mapping to the two-channel Kondo model, in which the dipole moment is mapped onto the operators with the scaling dimension Δm=1Δ_m=1Δm​=1, while the quadrupole moment onto the operators with another scaling dimension Δe=1/2Δ_e=1/2Δe​=1/2. Even for a fairly particle-hole asymmetric case with the Fermi-liquid ground state, the non-Fermi-liquid behavior has significant influences in electric and thermal properties.