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Physical Review B

Kondo Hamiltonian for a triplet impurity as a bridge between Fermi and non-Fermi-liquid physics

Mikito Koga, M. Matsumoto

Published Mar 14, 2008• 7 citations
Publisher

Abstract

A connection between competing Fermi and non-Fermi-liquid states is studied by a Kondo Hamiltonian for a triplet impurity, keeping cubic symmetry in mind. A uniaxial anisotropy stabilizes either state, corresponding to a local field in the two-channel Kondo model, which reflects on the local charge redistribution in the singlet-doublet configuration of the impurity. The numerical renormalization group and SU(2) group theoretical analyses show that the critical point is described by threefold degenerate terms of the two-channel Kondo Hamiltonian, one of which derives the non-Fermi-liquid state and the other two lead to the Fermi liquid.

DOI
10.1103/physrevb.77.094411

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A connection between competing Fermi and non-Fermi-liquid states is studied by a Kondo Hamiltonian for a triplet impurity, keeping cubic symmetry in mind. A uniaxial anisotropy stabilizes either state, corresponding to a local field in the two-channel Kondo model, which reflects on the local charge redistribution in the singlet-doublet configuration of the impurity. The numerical renormalization group and SU(2) group theoretical analyses show that the critical point is described by threefold degenerate terms of the two-channel Kondo Hamiltonian, one of which derives the non-Fermi-liquid state and the other two lead to the Fermi liquid.
We investigate the emergence of quantum critical points near a two-channel Kondo phase by evaluating an fff-electron entropy of a seven-orbital impurity Anderson model hybridized with three (Γ7Γ_7Γ7​ and Γ8Γ_8Γ8​) conduction bands with the use of a numerical renormalization group method First we consider the case of Pr3+^{3+}3+ ion, in which quadrupole two-channel Kondo effect is known to occur for the local Γ3Γ_3Γ3​ doublet state. When we control crystalline electric field (CEF) potentials so as to change the local CEF ground state from Γ3Γ_3Γ3​ doublet to Γ1Γ_1Γ1​ singlet or Γ5Γ_5Γ5​ triplet, we commonly observe a residual entropy of log⁡ϕ\log ϕlogϕ with the golden ratio ϕ=(1+5)/2ϕ=(1+\sqrt{5})/2ϕ=(1+5​)/2, which is equal to that for three-channel Kondo effect. This peculiar residual entropy is also observed for the case of Nd3+^{3+}3+ ion, in which magnetic two-channel Kondo phase is found to occur for the local Γ6Γ_6Γ6​ doublet state. We envisage a scenario that the quantum critical point characterized by log⁡ϕ\log ϕlogϕ generally appears between two-channel Kondo and Fermi-liquid phases.
The two-impurity Kondo model with particle-hole symmetry is known to have a critical point at comparable RKKY interaction and Kondo temperature. Using bosonization and controlled low energy projection, we identify the critical point and map it to a solvable resonant level model. The physics of the critical point is shown to be the same as the two-channel Kondo model. An effective Hamiltonian is derived from which the solutions for the critical point and surrounding Fermi-liquid phase follow immediately. The universality of the critical point is strongly supported by the overall excellent agreement with the numerical renormalization group and conformal field theory results.
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.
The two-channel Kondo Hamiltonian has recently been proposed as a description of several experimental systems. Our numerical renormalization-group treatment of a pair of magnetic impurities shows that at low temperatures, interimpurity interactions destabilize the ``marginal-Fermi-liquid'' behavior predicted by a single-impurity model. We find four stable zero-temperature regimes, three of which can be described by Fermi-liquid theory. The fourth possesses a complex many-body ground state and non-Fermi-liquid properties which are expected to be governed by nonuniversal critical exponents.
We have studied the quadrupolar Kondo effect for an impurity in cubic environment with taking account of a singlet excited state Γ1Γ_1Γ1​, which models a Pr3+\mathrm{Pr}^{3+}Pr3+ ion with a non-Kramers double ground state Γ3Γ_3Γ3​. We have used the numerical renormalization group approach and determined the phase diagram with varying quadrupole Kondo coupling JJJ and Γ1Γ_1Γ1​ excitation energy ΔΔΔ. Two phases are found and identified as local Fermi liquid and non-Fermi liquid. This non-Fermi liquid phase is characteristic to the two-channel Kondo impurity, and a similar phase diagram has been also found in other extended quadrupole models. We have analyzed in detail the JJJ-dependence of the Kondo temperature TKT_KTK​ near the phase boundary $J_c (Δ) $ and found a nice scaling behavior with an stretched exponential form TK∼δJαexp⁡(−const./δJ)T_K \sim δJ^α \exp (-\mathrm{const.}/\sqrt{δJ})TK​∼δJαexp(−const./δJ​) where δJ≡J−Jc(Δ)δJ \equiv J - J_c (Δ)δJ≡J−Jc​(Δ). This differs from the standard scaling form and indicates that one needs to consider renormalization of multiple coupling constants.
A Kondo model for three spin-one-half impurities placed at the vertices of an equilateral triangle is studied using the numerical renormalization-group method. The impurity spins can form two frustrated doublets which, because of the rotation symmetry, couple equally to the conduction band. Over a broad range of antiferromagnetic RKKY interaction strengths, this degeneracy drives the system to one of two novel, non-Fermi-liquid fixed points. Potential scattering is a relevant perturbation at one of these fixed points, but is marginal throughout the second regime.