# Effect of strong correlation on the study of renormalization group flow diagram for Kondo effect in a interacting quantum wire.

###### Abstract

We present the study of Kondo effect in an interacting quantum wire. We mainly emphasis the effect of strong electronic correlations in the study of renormalization group flow diagram and the stability analysis of fixed points for both magnetic and nonmagnetic impurities. We observe that the behavior of the system is either in the single channel or in the two channel Kondo effect depending on the initial values of coupling constants and strong correlations.

###### pacs:

: 73.63.Nm, 72.15.Qm, 71.10.Pm## I I. Introduction

One dimensional interacting electron systems are interesting in their own
right. In one dimension, interaction between the electrons has large
impact on the low-energy properties of the electron liquids. The basic
physics of electron liquid is changing from the Fermi liquid (FL) to
the Luttinger liquid (LL) physics. Such an one dimensional LL can be realized
in a very narrow quantum wire.
In this work, we are interested in the study of Kondo
effect in a quantum wire, where both magnetic and nonmagnetic impurities
are present. Kondo effect is one of the interesting phenomena of
correlated electronic systems since its discovery kon ; noz1 ; noz2 .
This effect arises
from the exchange interactions between an impurity spin and an electron
gas in three dimensions. In this case interaction changes from weak to strong
coupling as one decrease the temperature or energy scale. At T=0 the impurity
spin is completely screened by the conduction electrons and the basic physics
can be understood from the local FL kon ; noz1 ; noz2 . It was also observedthat when
the channel () exceeds twice of the impurity spin the system
is governed by the
physics of non-Fermi liquid theory andr ; aff .

In the conventional theory of Kondo effect, people have worked on an
independent
electron picture because in three dimensions the interacting electron system
can be described by the noninteracting quasiparticle (FL). However this scenario
is drastically changing in one dimension when the basic physics is govern by the LL.
The Kondo effect in Luttinger liquid was first discussed by Lee and Toner lee .
Furusaki
and Nagaosa nag have derived the scaling equations for the Kondo couplings in the
weak coupling regime by using the poor-man’s scaling method,
which preserves the
symmetry. They predicted that at low temperature the system is governed
by the strong coupling fixed point, where the impurity spin is screened completely.
They observed an important difference with conventional Kondo physics is that
Kondo coupling flows to the strong coupling regime for
both antiferromagnetic and
ferromagnetic coupling because the local backward scattering
potential is relevant
perturbation in the LL. But we shall see in our study that strong correlation
effect has not taken into account properly, i.e., the weak coupling RG flow
diagram is the same for noninteracting, repulsive and attractive LL.
The problem of a spin-1/2 magnetic impurity in a LL has also
been largely studied
in the literature john ; durga ; hur ; egg ; furu ; ravi ; meden .
It is well known that the repulsive
interaction due to nonmagnetic impurity breaks the wire at the impurity
site and treats the
residual tunneling through the barrier as a perturbation.
Here we study the two (dual) models that describe the impurity.
The plan of the work is
the following: In section II,
we present the weak coupling renormalization group flow
diagram with necessary physical analysis for a one dimensional interacting
electron system with only magnetic impurity. In sec. III, we present
weak-coupling RG
flow diagram of an one dimensional system with both magnetic and nonmagnetic impurities
are present. The section IV is devoted for conclusions.

## Ii II. Weak coupling renormalization group study of a quantum wire with only magnetic impurity

Here we consider a magnetic impurity of spin-1/2 at the origin of one dimensional interacting systems. It is well known that away from half-filling the basic physics of this type of system is governed by the spin sector of the Hamiltonian. We only consider the exchange coupling between the impurity spin and conduction electrons as follows

(1) | |||||

where and are respectively the forward and backward Kondo scattering coupling, is the pauli matrix. and are respectively the fermionic right and left mover field operators with spin . The field operators of right and left going electrons with spin are, , , where is the bosonic field. is the Klein factor that preserve the anticommutivity of fermionic field. During our continuum field theoretical calculations, we consider the following relations of the bosonic fields as

(2) | |||||

The scaling dimension for the forward scattering terms and backward scattering terms are respectively 1 and . So the backward scattering term is relevant for , where is the LL parameter of the charge sector. One can derive the renormalization group equation for one loop order by using poor-man’s scaling method. The RG equations are the following nag ; furu :

(3) |

These RG equations have only trivial fixed point, . We do the linear stability analysis to check the stability of these fixed points (FP). After the linear stability analysis RG equations reduce to

(4) |

where

and

. At the trivial fixed point, and

In Fig. 1, we present the RG flow diagram for the repulsive quantum wire, i.e,
. The Kondo couplings ( and ) are renormalized towards strong
coupling irrespective of the couplings are antiferromagnetic or ferromagnetic
Kondo couplings. This observation is in contrast with the three dimensional
Kondo effect. We observe from our study that the RG flow takes a large range of
initial conditions to the FP at (0,0). For all other initial conditions, we
see that there are two directions along which the Kondo
couplings flow to strong
coupling nag ; furu .

The one-loop RG equations suggest that apart from the trivial FP
(0,0), there are other three FPs, () = (+ ),
(+ ), (+ ). Here we want to discuss the FP (+ )
. It corresponds to the two channel Kondo problem. When , the spin and
the charge sectors are decoupled in the bosonized Hamiltonian. The
impurity spin is interacting with and only and
the Hamiltonian of the spin sector is equivalent to the
two-channel Kondo problem
with the right and left going electrons correspond to two channel.

. This explicit
study of the phase diagram was absent in the previous studies of
the Kondo effect in an one
dimensional interacting quantum wire and

In Fig. 2, we present the RG flow diagram for , i.e, the system of
attractive LL. We observe that the flow diagram and the two channel Kondo
regime is the same as repulsive LL (). It also reveals from our
study that noninteracting RG flow diagrams show the same behavior as the attractive and
repulsive LL. So we conclude that the behavior of the RG flow diagram is
the characteristic of the one dimensional system, interactions (different values
of LL parameter) have no effect.
This explicit study of strong electronic correlation on the study of Renormalization
Group flow diagram was absent in the previous study [7,12].
The RG equations of a quantum wire with a
magnetic impurity
are not sufficient to show up the strong correlation effects
because there is no breaking of quantum wire. Therefore one has to
consider the presence of strong nonmagnetic scatterer, which we will discuss
in the next section.

## Iii Renormalization group study of a quantum wire with magnetic impurity and nonmagnetic local potential

Here we present the RG study of an interacting wire with magnetic () and nonmagnetic () local potential. We study this part following Ref. 15. We consider the situation where the strength of local impurity potential is larger than the antiferromagnetic exchange . For this case one can first diagonalize the Hamiltonian with only local potential. It is well known that for repulsive interaction due to nonmagnetic impurity cutting the wire at the impurity site. Here we consider the tunneling as a perturbation. The Hamiltonian of the system is the following,

(5) | |||||

Where R and L correspond to the right and left side of the impurity, is the
impurity spin-1/2 operators and is the spin operator of the conduction
electrons. The first term represents the exchange interaction of the leads
with the impurity spin. The second term represents the tunneling process
with spin-flip. The third term presents the tunneling of electrons without
spin flip.

This model Hamiltonian represents the two channel Kondo model when and
are absent. The presence of and will introduce anisotropy
between two channels. In Ref. 15, the RG equation for this problem has derived.
We think that this derivation is not the complete one, i.e., there is one
more RG equation for the tunneling without spin flip. Now we present the
derivation of that RG equation very briefly.
One can write the boundary field operator fab as

(6) |

where

The scaling dimension of this term is

(7) |

Therefor the total RG equations are

(8) |

Here the FPs are the trivial ones, i.e., . We do linear stability analysis of the FPs to study the nature of the FPs. After the linear stability analysis the RG equations are reduced to

(9) |

where

and

.

At the trivial fixed point, ,

## Iv Conclusions

We have revisited the problem of magnetic and nonmagnetic impurity in a quantum wire. We have emphasized mainly the RG study of this problem. The RG flow diagram of the previous studies is the schematic one [7,12,15]. The Abelian bosonization study only reveals that wheather a coupling term is relevant or not. The RG flow diagram shows us explicitly for which initial values of coupling constants the systems flows to the single channel or two-channel Kondo problem.

Acknowledgments

The author would like to acknowledge, Prof. H. R. Krishnamurthy, Prof. Diptiman Sen, , Prof. P. Durganandini, and Dr. Ravi Chandra for useful discussions during the progress of the work and also Center for Condensed Matter Theory of the Physics Department of IISc for providing working space. The author finally acknowledge Dr. B. Mukhopadhay for reading the manuscript very critically.

## References

- (1)
- (2) J. Kondo, Prog. Theor. Phys. 32, 37 (1964).
- (3) P. Nozieres, J. Low Temp. Phys. 17, 31 (1974).
- (4) P. Nozieres and A. Blandin, J. Phys. (Paris) 41, 193 (1980).
- (5) N. Andrei and C. Destri, Phys. Rev. Lett. 52, 364 (1984); P. B. Wiegmann and A. M. Tsvelik, Z. Phys. B 54, 201 (1985).
- (6) I. Affleck and A. W. W. Ludwig, Phys. Rev. B 48, 7297 (1993).
- (7) D. H. Lee and J. Toner, Phys. Rev. Lett 69, 3378 (1992).
- (8) A. Furusaki and N. Nagaosa, Phys. Rev. Lett 72, 892 (1994).
- (9) P. Frojddh and H. Johannesson, Phys. Rev. Lett 75, 300 (1995).
- (10) P. Durganandini Phys. Rev. B 53, 8832 (1996).
- (11) K. Le. Hur, Phys. Rev. B 59, 11637 (1999).
- (12) S. Eggert, D. P. Gustafsson, and S. Rommer, Phys. Rev. Lett. 86, 516 (2001).
- (13) A. Furusaki, J. Phys. Soc. Jpn. 74, 73 (2005).
- (14) V. Ravi Chandra, S. Rao, and D. Sen, cond-mat/0510206.
- (15) S. Andergassen, T. Enss and V. Meden, Phys. Rev. B 73, 153308 (2006).
- (16) M. Fabrizio and A. O. Gogolin, Phys. Rev. B 51 17827 (1995).