We study a system of an impurity spin coupled to a junction of several Tomonaga-Luttinger liquids using a renormalization group scheme. For the decoupled S -matrix at the junction, there is a range of Kondo couplings which flow to a multi-channel fixed point for repulsive inter-electron interactions; this is associated with a characteristic temperature dependence of the spin-flip scatterings. If the junction is governed by the Griffiths S -matrix, the Kondo couplings flow to a strong-coupling fixed point where all the wires are decoupled.
We study a system consisting of a junction of
N wires, where the junction is characterized by a scalar
S-matrix, and an impurity spin is coupled to the electrons near the junction. The wires are modeled as weakly interacting Tomonaga-Luttinger liquids. We derive the renormalization group (RG) equations for the Kondo couplings of the spin to the electrons on different wires. We analyze the RG flows and fixed points for different values of the initial Kondo couplings and of the junction
S-matrix, such as the decoupled
S-matrix and the Griffiths (connected)
S-matrix. We find that the Kondo couplings flow either towards a ferromagnetic (FM) fixed point or towards large and antiferromagnetic (AFM) values in one of two ways. For the Griffiths
S-matrix, one of the strong coupling flows is towards a FM fixed point with decoupled wires; this is seen by a perturbative analysis. Thus if we start with a system of connected wires with an AFM coupling to the spin impurity, the flow at large distances is towards a system of disconnected wires at the FM fixed point. For the decoupled
S-matrix, the flow is either to a FM fixed point or to one of two strong coupling fixed points in which all the channels are strongly coupled to each other through the impurity spin. Strong interactions between the electrons with ${K}_{\ensuremath{\rho}}<N∕(N+2)$ can stabilize a multichannel fixed point in which the coupling between different channels goes to zero. We also study the temperature dependence of the conductance at the decoupled and Griffiths
S-matrices.
The Kondo effect in a repulsively interacting electron system (Tomonaga-Luttinger liquid) is studied. By using the poor man's scaling method it is shown that the Kondo coupling in this model flows to the strong-coupling regime not only for the antiferromagnetic but also for the ferromagnetic case. The ground state is governed by stable strong-coupling fixed points where the impurity spin is completely screened; the fixed-point Hamiltonian consists of two semi-infinite Tomonaga-Luttinger liquids and a spin singlet. Specific heat, susceptibility, and conductance are calculated for low temperatures.
To study the impurity potential effect to the Kondo problem in a Luttinger liquid, we propose an integrable model of two interacting half-chains coupled with a single magnetic impurity ferromagnetically. It is shown that the scalar potential effectively reconciles the spin dynamics at low temperatures. Generally, there is a competition between the Kondo coupling J and the impurity potential V. When the ferromagnetic Kondo coupling dominates over the impurity potential
(V<∣SJ∣), the Furusaki-Nagaosa many-body singlet can be perfectly realized. However, when the impurity potential dominates over the Kondo coupling
(V> ∣SJ∣), the fixed point predicted by Furusaki and Nagaosa is unstable and the system must flow to a weak-coupling fixed point. It is also found that the effective moment of the impurity measured from the susceptibility is considerably enlarged by the impurity potential. In addition, some quantum phase transitions driven by the impurity potential are found and the anomaly residual entropy is discussed.