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arXiv:cond-mat/9403039 · Physical review. B, Condensed matter

Microscopic calculations of the finite-size spectrum in the Kondo problem

Satoshi Fujimoto, Norio Kawakami, Sung-Kil Yang

Published Jul 1, 1994• 19 citations
arXivHTMLPDF

Abstract

The finite-size spectrum in the Kondo problem is obtained from the Bethe-ansatz solution of the exactly solved models. We investigate the Anderson model, the highly correlated SU(\ensuremath{\nu}) Anderson model, and the s-d exchange model. For all these models we find that the spectra exhibit the properties characteristic of the Fermi-liquid fixed point, and hence our microscopic calculations are in accordance with the results obtained by boundary conformal field theory with current algebra symmetry.

DOI
10.1103/physrevb.50.1046

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The finite-size spectrum in the Kondo problem is obtained from the Bethe-ansatz solution of the exactly solved models. We investigate the Anderson model, the highly correlated SU(\ensuremath{\nu}) Anderson model, and the s-d exchange model. For all these models we find that the spectra exhibit the properties characteristic of the Fermi-liquid fixed point, and hence our microscopic calculations are in accordance with the results obtained by boundary conformal field theory with current algebra symmetry.
We review some exact results on Kondo impurity systems derived from Bethe-ansatz solutions and boundary conformal field theory with particular emphasis on universal aspects of the phenomenon. The finite-size spectra characterizing the low-energy fixed point are computed from the Bethe-ansatz solutions of various models related to the Kondo problem. Using the finite-size scaling argument, we investigate their exact critical properties. We also discuss that a universal relation between the Kondo effect and the impurity effect in one-dimensional quantum systems usefully expedites our understanding of these different phenomena.
We solve the two-impurity, two-channel Kondo model using a combination of conformal invariance and bosonization techniques. In the odd-even symmetric case, the RKKY interaction is exactly marginal, resulting in a line of non-Fermi-liquid fixed points. Explicit formulas are given for the critical exponents and for the finite-size spectrum, which depend continuously on a single parameter. The marginal line spans a range of values of the RKKY coupling III which goes from the infinitely strong ferromagnetic point $I\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}+\ensuremath{\infty}$ to a finite antiferromagnetic critical value Imin<0{I}_{\mathrm{min}}<0Imin​<0 beyond which a Fermi liquid is recovered. When the odd-even symmetry is broken, the marginal line is unstable for ferromagnetic III, while for antiferromagnetic III it extends into a manifold of fixed points.
Exact critical exponents of the Green functions for pseudo-fermions and slave bosons in the SU(NNN) Anderson model with U→∞U\rightarrow\inftyU→∞ are obtained by using the Bethe ansatz solution and boundary conformal field theory. They are evaluated exactly for mixed valence systems and Kondo systems with crystalline fields. The results agree with the prediction of Menge and Müller-Hartmann, which coincide with those of the X-ray problem. Some implication of our results in one-dimensional chiral systems is also discussed.
We consider the single-channel Kondo problem with the Kondo coupling between a spin SSS impurity and conduction electrons with spin jjj. These problems arise as multicritical points in the parameter spaces of two-and higher-level tunneling systems and some impurity models of heavy fermion compounds. In contrast to the previous Bethe-ansatz conjectures, it turns out that the dynamics of the spin sector is the same as that of a spin-SSS impurity coupled to k(j)k(j)k(j) channels of spin-\textonehalf{} electrons with k(j)=2j(j+1)(2j+1)3k(j)=2\frac{j(j+1)(2j+1)}{3}k(j)=23j(j+1)(2j+1)​. As a result, for 2S<k(j)2S<k(j)2S<k(j), the system shows non-Fermi-liquid behavior with the same exponents for the thermodynamic quantities as those of the k(j)k(j)k(j)-channel Kondo problem. However, both the finite-size spectrum and the operator content are different due to the presence of the other sectors and can be obtained by conformal field theory techniques.
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.