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arXiv:cond-mat/0610631 · Physical Review B

Strong versus weak coupling duality and coupling dependence of the Kondo temperature in the two-channel Kondo model

Christian Kolf, Johann Kroha

Published Jan 26, 2007• 16 citations
arXivHTMLPDF

Abstract

We perform numerical renormalization group (NRG) as well as analytical calculations for the two-channel Kondo (2CK) model to obtain the dependence of the Kondo temperature ${T}_{K}$ on the dimensionless (bare) spin exchange coupling $g$ over the complete parameter range from $g⪡1$ to $g⪢1$. We show that there exists a duality between the regimes of small and large coupling. It is unique for the two-channel model and enables a mapping between the strong and the weak coupling cases via the identification $g\ensuremath{\leftrightarrow}3∕(2g)$, implying an exponential dependence of ${T}_{K}$ on $1∕g$ and $g$, respectively, in the two regimes. This agrees quantitatively with our NRG calculations where we extract ${T}_{K}(g)$ over the complete parameter range and obtain a nonmonotonic ${T}_{K}(g)$ dependence, strongly peaked at the 2CK fixed point coupling ${g}^{*}$. These results may be relevant for resolving the long-standing puzzle within the 2CK interpretation of certain random defect systems as to why no broad distribution of ${T}_{K}$ is observed in those systems.

DOI
10.1103/physrevb.75.045129

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We perform numerical renormalization group (NRG) as well as analytical calculations for the two-channel Kondo (2CK) model to obtain the dependence of the Kondo temperature TK{T}_{K}TK​ on the dimensionless (bare) spin exchange coupling ggg over the complete parameter range from g⪡1g⪡1g⪡1 to g⪢1g⪢1g⪢1. We show that there exists a duality between the regimes of small and large coupling. It is unique for the two-channel model and enables a mapping between the strong and the weak coupling cases via the identification g\ensuremath↔3∕(2g)g\ensuremath{\leftrightarrow}3∕(2g)g\ensuremath↔3∕(2g), implying an exponential dependence of TK{T}_{K}TK​ on 1∕g1∕g1∕g and ggg, respectively, in the two regimes. This agrees quantitatively with our NRG calculations where we extract TK(g){T}_{K}(g)TK​(g) over the complete parameter range and obtain a nonmonotonic TK(g){T}_{K}(g)TK​(g) dependence, strongly peaked at the 2CK fixed point coupling g∗{g}^{*}g∗. These results may be relevant for resolving the long-standing puzzle within the 2CK interpretation of certain random defect systems as to why no broad distribution of TK{T}_{K}TK​ is observed in those 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.
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.
We investigate theoretically the possibility of observing two-channel Kondo (2CK) physics in tunnel-coupled double quantum dots (TCDQDs), at both zero and finite magnetic fields; taking the two-impurity Anderson model as the basic TCDQD model, together with effective low-energy models arising from it by Schrieffer-Wolff transformations to second and third order in the tunnel couplings. The models are studied primarily using Wilson's numerical renormalization group. At zero field our basic conclusion is that while 2CK physics arises in principle provided the system is sufficiently strongly correlated, the temperature window over which it could be observed is much lower than is experimentally feasible. This finding disagrees with recent work on the problem, and we explain why. At finite field, we show that the quantum phase transition known to arise at zero field in the two-impurity Kondo model, with an essentially 2CK quantum critical point, persists at finite fields. This raises the prospect of access to 2CK physics by tuning a magnetic field, although preliminary investigation suggests this to be even less feasible than at zero field.
We investigate the low-temperature dynamics of single electron boxes and transistors close to their degeneracy point using renormalization-group methods. We show that intermode scattering is a relevant perturbation and always drives the system to the two-channel Kondo fixed point, where the two channels correspond to the real spins of the conduction electrons. However, the crossover temperature T∗,{T}^{*},T∗, below which Matveev's two-channel Kondo scenario [K.A. Matveev, Phys. Rev. B 51, 1743 (1995)] develops decreases exponentially with the number of conduction modes in the tunneling junctions and is extremely small in most cases. Above T∗{T}^{*}T∗ the ``infinite channel model'' of G. Falci, G. Sch\"on, and G.T. Zimany, Phys. Rev. Lett. 74, 3257 (1995) turns out to be a rather good approximation. We discuss the experimental limitations and suggest an experimental setup to observe the multichannel Kondo behavior.
We analyze the nonequilibrium Kondo model at finite voltage and temperature by using a new formulation of the real-time renormalization group method with the Laplace variable as the flow parameter. We evaluate the energy-dependent spin relaxation rate and nonlinear conductance, and derive an approximate form for the universal line shape for the latter in the whole crossover regime from weak to strong coupling. The results are shown to agree well with exact methods in equilibrium, Fermi-liquid theory, weak-coupling expansions, and recent experiments. For the transient spin dynamics we find a universal exponential decay in the long-time limit along with a truncation-dependent pre-exponential power law. For multichannel models a pure power-law decay typical for non-Fermi-liquid behavior is predicted.
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.