Low Temperature Behavior of the Kondo Effect in Tomonaga-Luttinger Liquid

Using the bosonization method, we study the low temperature behavior of the Kondo effect in the Tomonaga-Luttinger liquid and clearly show that the power law temperature dependence of the impurity susceptibility is completely determined by the repulsive electron-electron interaction existing in the total spin channel and is independent of the electron-electron interaction existing in the charge channels. 78.70.Dm, 79.60.Jv, 72.10.Fk, 71.45.-d Typeset using REVTEX 1 Recently, the quantum impurity scattering of the Tomonaga-Luttinger(TL) liquid has been extensively studied by using different techniques[ [1][18]]. However, there is some controversy on the treatment of backward scattering of the conduction electrons on a quantum impurity or impurity-like hole in the valence band, although we all agree that the backward scattering drastically changes the behavior of a TL-liquid. The main difficulty is that we have not a reasonable non-perturbation technique to treat strongly correlated systems such as in the high energy physics and condensed matter physics. Another interesting problem is a quantum magnetic impurity scattering of the TL-liquid (Kondo model in onedimensional electronic system). Although it is extensively studied by many authors[ [20][23]], the low energy and low temperature properties of the Kondo effect in the TL-liquid still remain an open problem because of the strong coupling between the impurity spin and the conduction electrons in the low energy and low temperature limit. In this paper, using the bosonization method, we give a detail study on the low temperature behavior of the Kondo effect in the TL-liquid and first time clearly show that the power law non-Fermi liquid behavior of the impurity susceptibility completely depends upon the electron-electron interaction existing in the total spin channel and is independent of the electron-electron interaction existing in the charge channels. For a free electron system, g = 1 (g is a dimensionless coupling strength parameter, g = 1 corresponding to the free electron gas), the impurity part shows a Fermi liquid behavior. For a weak electronelectron interaction, gc < g < 1, gc is defined as satisfying relation: (1 − g c ) = 2gc, the impurity susceptibility has a power law temperature dependence. For a strong repulsive electron-electron interaction, g ≤ gc, the impurity susceptibility satisfies the Curie law and the impurity fermion has a free fermion Green function. Therefore, for the repulsive electronelectron interaction case, the impurity susceptibility shows a non-Fermi liquid behavior. This surprising behavior of the impurity spin in the TL-liquid is completely determined by the strongly coupling fixed point Hamiltonian. However, in the strong coupling limit, we may have either the Fermi liquid fixed point or the non-Fermi liquid fixed point which completely depends upon the electron-electron interaction existing in the total spin channel. For an 2 one-dimensional free electron system, g = 1, the system reduces into an anisotropic twochannel Kondo model, the backward scattering potential provides the channel anisotropy which completely destroys the non-Fermi liquid behavior of the isotropic two-channel Kondo model and makes the system have the Fermi liquid behavior. We choose the following Hamiltonian to describe the Kondo effect in an one-dimensional interacting electronic system