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Physical review. B, Condensed matter

Stability of the fixed point of the two-channel Kondo Hamiltonian

H.-B. Pang, D. L. Cox

Published Nov 1, 1991• 90 citations
Publisher

Abstract

We have performed full-scale numerical-renormalization-group calculations on the anisotropic, two-channel, impurity spin s=1/2 Kondo Hamiltonian. To allow for anisotropy we have worked in a basis of total axial charge ${\mathit{j}}_{\mathit{a}}$ and ${\mathit{j}}_{\mathit{b}}$ (a and b index the channels) as well as total ${\mathit{S}}_{\mathit{z}}$. In the isotropic limit, our results agree well with the previous work of Cragg, Lloyd, and Nozieres. The most important results are (1) confirmation of the approach to a nontrivial fixed point from weak coupling, (2) the stability of the fixed point against exchange anisotropy, and (3) the destabilization of the fixed point by the application of a magnetic field or the lifting of exchange symmetry between two channels. We find the crossover temperature scales with the square of the exchange splitting.

DOI
10.1103/physrevb.44.9454

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We have performed full-scale numerical-renormalization-group calculations on the anisotropic, two-channel, impurity spin s=1/2 Kondo Hamiltonian. To allow for anisotropy we have worked in a basis of total axial charge ja{\mathit{j}}_{\mathit{a}}ja​ and jb{\mathit{j}}_{\mathit{b}}jb​ (a and b index the channels) as well as total Sz{\mathit{S}}_{\mathit{z}}Sz​. In the isotropic limit, our results agree well with the previous work of Cragg, Lloyd, and Nozieres. The most important results are (1) confirmation of the approach to a nontrivial fixed point from weak coupling, (2) the stability of the fixed point against exchange anisotropy, and (3) the destabilization of the fixed point by the application of a magnetic field or the lifting of exchange symmetry between two channels. We find the crossover temperature scales with the square of the exchange splitting.
The phase diagram of the multichannel Kondo Hamiltonian with an XXZXXZXXZ spin-exchange anisotropy is revisited, revealing a far richer fixed-point structure than previously appreciated. For a spin-12\frac{1}{2}21​ impurity and k>2k>2k>2 conduction-electron channels, a second ferromagneticlike domain is found deep inside the antiferromagnetic regime. The new domain extends above a (typically large) critical longitudinal coupling Jz∗>0{J}_{z}^{*}>0Jz∗​>0 and is separated from the antiferromagnetic domain by a second Kosterlitz-Thouless line. A similar line of stable ferromagneticlike fixed points with a residual isospin-12\frac{1}{2}21​ local moment is shown to exist for large ${J}_{z}⪢\ensuremath{\mid}{J}_{\ensuremath{\perp}}\ensuremath{\mid}>0$ and arbitrary kkk and sss obeying $\ensuremath{\mid}k\ensuremath{-}2s\ensuremath{\mid}>1$. Here, Jz{J}_{z}Jz​ is the longitudinal spin-exchange coupling, ${J}_{\ensuremath{\perp}}$ is the transverse coupling, and sss is the impurity spin. Near the free-impurity fixed point, spin-exchange anisotropy generates a highly relevant term for s>1∕2s>1∕2s>1∕2 and arbitrary kkk. Depending on the sign of ${J}_{z}^{2}\ensuremath{-}{J}_{\ensuremath{\perp}}^{2}$ and the parity of 2s2s2s, the system flows either to a conventional Fermi liquid with no residual degeneracy or to a kkk-channel, spin-12\frac{1}{2}21​ Kondo effect or to a line of ferromagneticlike fixed points with a residual isospin-12\frac{1}{2}21​ local moment. These results are obtained through a combination of perturbative renormalization-group techniques, Abelian bosonization, a strong-coupling expansion in 1∕Jz1∕{J}_{z}1∕Jz​, and explicit numerical renormalization-group calculations.
The multichannel Kondo model exhibits non-Fermi-liquid behavior in the overscreened case, when the number of channels, k, is greater than twice the size of the impurity spin, s. We show that, for overscreening, exchange anisotropy is irrelevant at the low-temperature fixed point for s=1/2 or s=k/2-1/2, but relevant for all other values of s. However, an external field or channel asymmetry is relevant, producing crossover to Fermi-liquid fixed points with different phase shifts for each spin in the first case and for each channel in the second. These results are elucidated by explicit comparison of analytic finite-size spectra derived from conformal field theory with those obtained from numerical renormalization-group calculations. The relevance of the results (for k=2 and s=1/2) to the quadrupolar Kondo Hamiltonian (which has been proposed as a model for many uranium-based heavy-fermion materials) will be briefly discussed. Also, a larger, a/Isymplectic symmetry Sp(2k) is shown to be present in the k-channel Kondo model.
Using numerical renormalization-group techniques, we study static and dynamic properties of a family of single-channel Kondo impurity models with axial magnetic anisotropy DSz2D{S}_{z}^{2}DSz2​ terms; such models are appropriate to describe magnetic impurity atoms adsorbed on nonmagnetic surfaces, which may exhibit surface Kondo effect. We show that for positive anisotropy DDD and for any spin SSS, the systems behave at low temperatures as regular Fermi liquids with fully compensated impurity spin. The approach to the stable fixed point depends on the value of the spin SSS and on the ratio D/TK(0)D/{T}_{K}^{(0)}D/TK(0)​, where TK(0){T}_{K}^{(0)}TK(0)​ is the Kondo temperature in the absence of the anisotropy. For S=1S=1S=1, the screening occurs in two stages if D<TK(0)D<{T}_{K}^{(0)}D<TK(0)​; the second Kondo temperature is exponentially reduced in this case. More generally, there is an effective spin-1/2 Kondo effect for any integer SSS if D<TK(0)D<{T}_{K}^{(0)}D<TK(0)​ and for any half-integer SSS if D>TK(0)D>{T}_{K}^{(0)}D>TK(0)​. For negative anisotropy DDD, the system is a non-Fermi liquid with residual anisotropic exchange interaction. However, the presence of transverse magnetic anisotropy $E({S}_{x}^{2}\ensuremath{-}{S}_{y}^{2})$ restores Fermi-liquid behavior in real systems.
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 Delta0Delta_0Delta0​. When immersed in a metal, electronic angular momentum scattering induces transitions between any two of these levels (couplings JJJ), 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)=kBln2S(T_K < T < Delta_0) = k_B ln 2S(TK​<T<Delta0​)=kB​ln2 between the high-TTT value, S(TggDelta0)=kBln3S(T gg Delta_0) = k_B ln 3S(TggDelta0​)=kB​ln3, and the 2CK ground state value, S(0)=kBlnsqrt2S(0) = k_B lnsqrt{2}S(0)=kB​lnsqrt2. 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−Delta0J-Delta_0J−Delta0​ plane. The Kondo temperature TKT_KTK​ shows non-monotonic JJJ-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)=kBln2S(T_K < T < Delta_0) = k_B ln 2S(TK​<T<Delta0​)=kB​ln2 between high-TTT value, S(TggDelta0)=kBln3S(T gg Delta_0) = k_B ln 3S(TggDelta0​)=kB​ln3, and the single-channel Kondo ground state value, S(0)=kBln1S(0) = k_B ln 1S(0)=kB​ln1.
Numerical renormalization-group results on entropy of the anisotropic two-channel Kondo model with the band-width cutoff (DDD) in the presence of a magnetic field (hhh) are obtained to determine crossover temperature from the non-Fermi liquid to Fermi liquid fixed point. It is found that the crossover temperature (TxT_{\rm x}Tx​) is given by Tx≡rTK∼D(ΔJ/Jav)2e−1/JavT_{\rm x} \equiv {r} T_{\rm K} \sim D(ΔJ/J_{\rm av})^2 e^{-1/J_{\rm av}}Tx​≡rTK​∼D(ΔJ/Jav​)2e−1/Jav​ when $(h /T_{\rm K})^2 \ll r \ll 1 $, where TKT_{\rm K}TK​, JavJ_{\rm av}Jav​ and ΔJΔJΔJ are the Kondo temperature, the average and difference of the exchange coupling constants, respectively. This result indicates that non-Fermi liquid behavior can be seen even if ΔJ>TKΔJ > T_{\rm K}ΔJ>TK​. Robust similarities of the crossover behavior in the region around the non-Fermi liquid critical point to that of the two-impurity Kondo model are also discussed.
Dynamical correlations of various local operators are studied in the spin-half two-channel Kondo (2CK) model in the presence of channel anisotropy or external magnetic field. A conformal field theory--based scaling approach is used to predict the analytic properties of various spectral functions in the vicinity of the two-channel Kondo fixed point. These analytical results compare well with highly accurate density-matrix numerical renormalization-group results. The universal crossover functions interpolating between channel-anisotropy- or magnetic-field-induced Fermi-liquid regimes and the two-channel Kondo, non-Fermi-liquid regimes are determined numerically. The boundaries of the real 2CK scaling regime are found to be rather restricted and to depend both on the type of the perturbation and on the specific operator whose correlation function is studied. In a small magnetic field, a universal resonance is observed in the local fermion's spectral function. The dominant superconducting instability appears in the composite superconducting channel.