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arXiv:cond-mat/9609159 · Europhysics Letters (EPL)

Renormalization-group study of a magnetic impurity in a Luttinger liquid

A. Schiller, Kevin Ingersent

Published Sep 15, 1997• 12 citations
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Abstract

A generalized Anderson model for a magnetic impurity in an interacting one-dimensional electron gas is studied via a mapping onto a classical Coulomb gas. For weak potential scattering, the local-moment parameter regime expands as repulsive bulk interactions become stronger, but the Kondo scale for the quenching of the impurity moment varies non-monotonically. There also exist two regimes dominated by backward potential scattering: one in which the impurity is non-magnetic, and another in which an unquenched local moment survives down to very low temperatures.

DOI
10.1209/epl/i1997-00408-4

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A generalized Anderson model for a magnetic impurity in an interacting one-dimensional electron gas is studied via a mapping onto a classical Coulomb gas. For weak potential scattering, the local-moment parameter regime expands as repulsive bulk interactions become stronger, but the Kondo scale for the quenching of the impurity moment varies non-monotonically. There also exist two regimes dominated by backward potential scattering: one in which the impurity is non-magnetic, and another in which an unquenched local moment survives down to very low temperatures.
We study the low-temperature properties of a spin-1/2 magnetic impurity coupled to a one-dimensional interacting electron system. Using a formalism by Affleck and Ludwig, with a scale invariant boundary condition replacing the impurity, we exploit boundary conformal field theory to deduce the impurity thermal and magnetic response. In the case of only forward electron scattering off the impurity, we predict the same critical scaling as for the two-channel Kondo effect for noninteracting electrons, but with a different Wilson ratio. Backward electron scattering off the impurity destabilizes this behavior and drives the system to a new fixed point. In the case of equal amplitudes for forward and backward scattering (Kondo interaction), we show that there are only two types of scaling behaviors consistent with the symmetries of the problem: either a local Fermi liquid or a critical theory with an anomalous specific heat. The latter case agrees with a recent poor-man's scaling result proposed by Furusaki and Nagaosa. \textcopyright{} 1996 The American Physical Society.
To study the impurity potential effect to the Kondo problem in a Luttinger liquid, we introduce a model of ferromagnetic single-impurity Kondo problem in a multi-band one-dimensional δ-potential Fermi gas, which is exactly solvable for a special Kondo coupling constant via Bethe ansatz. It is found that the boundary potential modifies the local moment seriously due to the charge-spin coupling and the strong backward scattering off the impurity, and thus strongly affects the fixed point of the system. Generally, their is a residual entropy for the ground state which depends on the strength of the boundary potential. In addition, the tunneling effect through the impurity and the breakdown of the open boundary induced by the impurity are also discussed.
We study the dynamics of a heavy particle of mass M moving in a one-dimensional repulsively interacting Fermi gas. The Fermi gas is described using the Luttinger model and bosonization. By transforming to a frame comoving with the heavy particle, we map the model onto a generalized ``quantum impurity problem.'' A renormalization-group calculation reveals a crossover from strong to weak coupling upon scaling down in temperature. Above the crossover temperature scale T∗{\mathit{T}}^{\mathrm{*}}T∗=(m/M)EF{\mathit{E}}_{\mathit{F}}EF​, the particle's mobility \ensuremath{\mu} is found to be (roughly) temperature independent and proportional to the dimensionless conductance g, characterizing the one-dimensional Luttinger liquid. Here m(\ensuremath{\ll}M) is the fermion mass, and EF{\mathit{E}}_{\mathit{F}}EF​ the Fermi energy. Below T∗{\mathit{T}}^{\mathrm{*}}T∗, in the weak-coupling regime, the mobility grows and diverges as \ensuremath{\mu}(T)\ensuremath{\sim}${\mathit{T}}^{\mathrm{\ensuremath{-}}4}$ in the T\ensuremath{\rightarrow}0 limit. \textcopyright{} 1996 The American Physical Society.
A quantum Monte Carlo simulation method has been developed and applied to study the critical behavior of a single Kondo impurity in a Luttinger liquid. This numerically exact method has no finite-size limitations and allows to simulate the whole temperature range. Focusing on the impurity magnetic susceptibility, we determine the scaling functions, in particular for temperatures well below the Kondo temperature. In the absence of elastic potential scattering, we find Fermi-liquid behavior for strong electron-electron interactions, gc<12{g}_{c}<\frac{1}{2}gc​<21​, and anomalous power laws for 12<gc<1\frac{1}{2}<{g}_{c}<121​<gc​<1, where gc{g}_{c}gc​ is the correlation parameter of the Luttinger liquid. These findings resolve a recent controversy. If elastic potential scattering is present, we find a logarithmically divergent impurity susceptibility at gc<12{g}_{c}<\frac{1}{2}gc​<21​ that can be rationalized in terms of the two-channel Kondo model.