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Conformal Field Theory Approach to the Kondo Effect

Ian Affleck

TL;DR

This work develops a boundary conformal field theory framework for the Kondo problem, tying traditional RG and Fermi-liquid analyses to a CFT description that uses boundary states and fusion to capture impurity effects. By decomposing the problem into charge, spin, and flavour sectors and employing Cardy’s boundary CFT and Verlinde fusion, it yields precise predictions for finite-size spectra, impurity entropy, and boundary Green's functions, including non-Fermi-liquid behavior in the multi-channel (overscreened) case. The approach explains how the infrared fixed point can be obtained via fusion with spin-s primaries, computes universal scaling forms for impurity thermodynamics (C_imp, χ_imp, R_W) and resistivity at T>0, and provides quantitative links to experimental observables in two-channel Kondo systems. Overall, the boundary CFT method offers a powerful, exacting framework for understanding quantum impurity problems beyond the single-channel Fermi liquid paradigm, with broad implications for non-Fermi-liquid physics in condensed matter and related fields.

Abstract

Recently, a new approach, based on boundary conformal field theory, has been applied to a variety of quantum impurity problems in condensed matter and particle physics. A particularly enlightening example is the multi-channel Kondo problem. In this review some earlier approaches to the Kondo problem are discussed, the needed material on boundary conformal field theory is developed and then this new method is applied to the multi-channel Kondo problem.

Conformal Field Theory Approach to the Kondo Effect

TL;DR

This work develops a boundary conformal field theory framework for the Kondo problem, tying traditional RG and Fermi-liquid analyses to a CFT description that uses boundary states and fusion to capture impurity effects. By decomposing the problem into charge, spin, and flavour sectors and employing Cardy’s boundary CFT and Verlinde fusion, it yields precise predictions for finite-size spectra, impurity entropy, and boundary Green's functions, including non-Fermi-liquid behavior in the multi-channel (overscreened) case. The approach explains how the infrared fixed point can be obtained via fusion with spin-s primaries, computes universal scaling forms for impurity thermodynamics (C_imp, χ_imp, R_W) and resistivity at T>0, and provides quantitative links to experimental observables in two-channel Kondo systems. Overall, the boundary CFT method offers a powerful, exacting framework for understanding quantum impurity problems beyond the single-channel Fermi liquid paradigm, with broad implications for non-Fermi-liquid physics in condensed matter and related fields.

Abstract

Recently, a new approach, based on boundary conformal field theory, has been applied to a variety of quantum impurity problems in condensed matter and particle physics. A particularly enlightening example is the multi-channel Kondo problem. In this review some earlier approaches to the Kondo problem are discussed, the needed material on boundary conformal field theory is developed and then this new method is applied to the multi-channel Kondo problem.

Paper Structure

This paper contains 18 sections, 269 equations, 27 figures, 3 tables.

Table of Contents

  1. Renormalization Group and Fermi Liquid Approaches to the Kondo Effect
  2. Introduction to the Kondo Effect
  3. Renormalization Group
  4. Mapping to a One-Dimensional Model
  5. Fermi Liquid Approach at Low T
  6. Conformal Field Theory Techniques
  7. Conformal Field Theory Approach to The Kondo Effect
  8. Leading Irrelevant Operator:Specific Heat, Susceptibility, Wilson Ratio, Resistivity at $T>0$
  9. Multi-Channel Kondo Effect
  10. Large-k Limit
  11. Current Algebra Approach
  12. Boundary Conformal Field Theory
  13. Boundary Conformal Field Theory Results on the Multi-Channel Kondo Effect
  14. Fusion and the Finite-Size Spectrum
  15. Impurity Entropy
  16. ...and 3 more sections

Figures (27)

  • Figure 1: Reduction of the cut-off from $D$ to $D'$.
  • Figure 2: Feynman diagrams contributing to renormalization of the Kondo coupling constant to third order.
  • Figure 3: RG flow of the Kondo coupling.
  • Figure 4: Reflecting the left-movers to the negative axis.
  • Figure 5: Free fermion energy levels with antiperiodic boundary conditions.
  • ...and 22 more figures