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CRC Press eBooks

Exact Results in the Kondo Problem. II. Scaling Theory, Qualitatively Correct Solution and Some New Results on One-Dimensional Classical Statistical Models

P.W. Anderson, G. Yuval, Donald R. Hamann

Published Mar 9, 2018• 6 citations
Publisher

Abstract

The simplest Kondo problem is treated exactly in the ferromagnetic case, and given exact bounds for the relevant physical properties in the antiferromagnetic case, by use of a Bcaling technique on an asymptotically exact expression for the ground-state properties given earlier. The theory also solves the n = 2 case of the one-dimensional Ising problem. The ferromagnetic case has a finite spin, while the antiferromagnetic case has no truly singular T → 0 properties (e.g., it has finite χ).

DOI
10.4324/9780429494116-21

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The simplest Kondo problem is treated exactly in the ferromagnetic case, and given exact bounds for the relevant physical properties in the antiferromagnetic case, by use of a Bcaling technique on an asymptotically exact expression for the ground-state properties given earlier. The theory also solves the n = 2 case of the one-dimensional Ising problem. The ferromagnetic case has a finite spin, while the antiferromagnetic case has no truly singular T → 0 properties (e.g., it has finite χ).
To specify the difference between the one-dimensional Kondo problem and the three-dimensional Kondo problem, we consider a model which describes the interaction between one-dimensional free electrons and a magnetic impurity. Bethe ansatz exact results and scaling analysis show that if J F > -| J B |, the Kondo screening occurs. There is no ferromagnetic Kondo screening in such a system but the long range antiferromagnetic Kondo coupling is relevant. As soon as the Coulomb interaction is introduced, the crossovers between the scaling trajectories in the parameter region J F < -| J B | and the trajectories in other parameter regions arise and ferromagnetic Kondo screening may occur.
We propose an integrable Kondo problem in a one-dimensional $t\ensuremath{-}J$ model. With the open boundary condition of the wave functions at the impurity sites, the model can be exactly solved via Bethe ansatz for a set of JL,R{J}_{L,R}JL,R​ (Kondo coupling constants) and VL,R{V}_{L,R}VL,R​ (impurity potentials) parametrized by a single parameter ccc. The integrable value of JL,R{J}_{L,R}JL,R​ runs from negative infinity to positive infinity, which allows us to study both the ferromagnetic Kondo problem and the antiferromagnetic Kondo problem in a strongly correlated electron system. Generally, there is a residual entropy for the ground state, which indicates a typical non-Fermi liquid behavior.
The equilibrium Kondo problem is solved using a spin-boson representation of the partition function. For antiferromagnetic coupling, a canonical transformation removes infrared singularities in the energy and magnetization series. At T=0T=0T=0 the moment vanishes, the energy is nonanalytic in ${J}_{\ensuremath{\perp}}$, the magnetization saturates at a field HS{H}_{S}HS​ of order ${(\frac{{{J}_{\ensuremath{\perp}}}^{2}}{{J}_{\ensuremath{\parallel}}})}^{\frac{1}{2{J}_{\ensuremath{\parallel}}}}$, and the susceptibility $\ensuremath{\chi}$ is proportional to ${{H}_{S}}^{\ensuremath{-}1}$. ${J}_{\ensuremath{\parallel}}$ and ${J}_{\ensuremath{\perp}}$ are longitudinal and transverse exchange constants.) For ferromagnetic coupling the free moment is only slightly renormalized.
It is well known that the free electron Kondo problem can be described by a one-dimensional (1D) model because only the s-wave part of the electronic wavefunction is affected by the Kondo coupling. Moreover, since only the spin degrees of freedom are involved in the Kondo interaction, and due to spin–charge separation in 1D, the universal low energy long distance physics of the Kondo model also arises when a magnetic impurity is coupled to the end of a gapless antiferromagnetic J 1 –J 2 spin- chain, where J 1 ( J 2 ) is the (next) nearest neighbor coupling. Experimental realizations of such spin chain models are possible and, using various analytical and numerical techniques, we present a detailed and quantitative comparison between the usual free electron Kondo model and such spin chain versions of the Kondo problem. For the gapless J 1 –J 2 spin chain two cases are studied, with zero next nearest neighbor coupling, J 2 = 0, and with a critical second-neighbor coupling, J 2 = J 2 c . We first focus on the spin chain impurity model with a critical second-neighbor antiferromagnetic exchange where a bulk marginal coupling present in the spin chain model for J 2 < J 2 c vanishes . There, the usual Kondo physics is recovered in the spin chain model in the low energy regime (up to negligible corrections, dropping as powers of inverse length or energy). At J 2 c the spin chain model is not exactly solvable and we demonstrate the equivalence to the Kondo problem by comparing density matrix renormalization group calculations on the frustrated spin chain model with exact Bethe ansatz calculations of the electronic Kondo problem. We then analyze the nearest neighbor model ( J 2 = 0) where a new kind of Kondo effect occurs due to the presence of the bulk marginal coupling. This marginal coupling slightly alters the β function for the Kondo coupling, leading to a slower variation of the Kondo temperature, T K , with the bare Kondo coupling. In their exact Bethe ansatz solution of this spin chain impurity model ( J 2 = 0), Frahm and Zvyagin noted this relation as well as the connection to the Kondo problem. Here, by numerically solving the Bethe ansatz equations we provide further evidence for the connection to Kondo physics and in addition we present low temperature quantum Monte Carlo results for the impurity susceptibility that further support this connection.
We reinvestigate the large degeneracy solution of the multichannel Kondo problem, and show how in the universal regime the complicated integral equations simplifying the problem can be mapped onto a first-order differential equation. This leads to an explicit expression for the full zero-temperature scaling functions at---and away from---the intermediate non-Fermi-liquid fixed point, providing complete analytic information on the universal low---and intermediate---energy properties of the model. These results also apply to the widely used noncrossing approximation of the Anderson model, taken in the Kondo regime. An extension of this formalism for studying finite-temperature effects is also proposed and offers a simple approach to solve models of strongly correlated electrons with relevance to the physics of heavy fermion compounds.