We consider an atom in a double-well potential [parametrized by a two-level system (TLS)] interacting with screened conduction electrons. The tunneling of the TLS is assisted by the Fermi gas. Close to the strong-coupling fixed point the renormalization-group equations for the noncommutative model in the case of spinless fermions, but an arbitrary number of relevant orbital channels, lead to the n-channel Kondo problem with S=1/2. By solving the thermodynamic Bethe-ansatz equations for the n-channel Kondo problem numerically in the presence of a magnetic field, we discuss the low-temperature properties of the TLS close to the fixed point. The Zeeman splitting corresponds to the energy difference between the two positions of the tunneling atom. The atom is not localized at one of the potential minima. The susceptibility diverges as H\ensuremath{\rightarrow}0 and T\ensuremath{\rightarrow}0, indicating that the symmetric TLS is unstable against a local lattice deformation via coupling to phonons. The lattice distortion disappears above a critical temperature
Tc. The ground-state equilibrium situation of the TLS corresponds to H\ensuremath{\ne}0 and a Fermi-liquid picture applies. The values for the specific heat become very large as H\ensuremath{\rightarrow}0. For small fields the specific heat shows a double-peak structure, which is particularly pronounced for n=2.
There are strong recent indications that the Kondo phenomena play important role in the electronic transport and dephasing mechanism for some of the experimentally investigated metallic samples. The spin and the orbital Kondo problem with two-level systems provide very similar behavior above the Kondo temperature. Below the Kondo temperature the behavior is drastically different as in the latter case the entropy remains non zero. In the following several results derived earlier by different methods are simply reproduced by the time-ordered diagram technique in the leading logarithmic approximation valid above the Kondo temperature.
The article reviews recent results for the low energy physics of fast tunneling centers in metallic environments. For strong enough couplings to the environment these tunneling centers display an orbital Kondo effect and give rise to a non-Fermi-liquid behavior. This latter property is explained by establishing a mapping of the tunneling center model to the multichannel Kondo model via the renormalization group transformation combined with a 1/Nf expansion. The case of M-state systems, the role of the splittings and the present experimental situation are also discussed.
We review analytical and numerical results derived from the Bethe ansatz solution of the n-channel Kondo model of arbitrary spin S as a function of temperature, external field, impurity spin S and the number of channels n. Three situations have to be distinguished: (i) If n = 2S the conduction electrons exactly compensate the impurity spin into a singlet at low temperatures, (ii) if n < 2S the impurity spin is only partially compensated (undercompensated), and (iii) if n > 2S the impurity spin is said to be overcompensated giving rise to critical behaviour. The results are discussed in the context of magnetic impurities, e.g. Fe, Cr and Tm in simple metals, the quadrupolar Kondo effect, an impurity spin embedded in the Takhtajan-Babujian Heisenberg model and electron assisted-tunnelling of an atom in a double-well potential.
The interaction between an atom moving in a model double-well potential and the conduction electrons is treated using renormalization-group methods in next-to-leading logarithmic order. A large number of excited states is taken into account and the Kondo temperature
TK is computed as a function of barrier parameters. We find that for special parameters
TK can be close to 1 K and it can be of the same order of magnitude as the renormalized splitting $\ensuremath{\Delta}.$ However, in the perturbative regime we always find that ${T}_{K}\ensuremath{\lesssim}\ensuremath{\Delta}$ with ${T}_{K}\ensuremath{\lesssim}1\mathrm{K}$ [I. L. Aleiner et al., Phys. Rev. Lett. 86, 2629 (2001)]. We also find that $\ensuremath{\Delta}$ remains unrenormalized at energies above the Debye frequency, ${\ensuremath{\omega}}_{\mathrm{Debye}}.$
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.