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Lectures on Non Perturbative Field Theory and Quantum Impurity Problems: Part II

H. Saleur

TL;DR

The notes develop and apply the massless form-factor program to integrable quantum field theories with impurities, extending the standard massive framework to boundary problems and massless scattering. By constructing explicit two-particle bootstrap relations, minimal form-factors, and boundary states, the author shows how to compute current and vertex-operator correlators with controlled accuracy, and then translates these correlators into observable quantities such as AC conductance in fractional quantum Hall edge states and spin dynamics in the anisotropic Kondo model. The approach reveals rapid convergence of the form-factor series, clarifies UV/IR behavior, and provides practical, quantitatively accurate predictions across a range of strongly correlated impurity problems, including Friedel oscillations and Shiba-type relations. While powerful, the method faces technical hurdles near isotropic limits, yet remains broadly applicable to a variety of impurity-related phenomena in integrable field theories.

Abstract

These are notes of lectures given at The NATO Advanced Study Institute/EC Summer School on ``New Theoretical Approaches to Strongly Correlated Systems'' (Newton Institute, April 2000). They are a sequel to the notes I wrote two years ago for the Summer School ``Topological Aspects of Low Dimensional Systems'', (Les Houches, July 1998). In this second part, I review the form-factors technique and its extension to massless quantum field theories. I then discuss the calculation of correlators in integrable quantum impurity problems, with special emphasis on point contact tunneling in the fractional quantum Hall effect, and the two-state problem of dissipative quantum mechanics.

Lectures on Non Perturbative Field Theory and Quantum Impurity Problems: Part II

TL;DR

The notes develop and apply the massless form-factor program to integrable quantum field theories with impurities, extending the standard massive framework to boundary problems and massless scattering. By constructing explicit two-particle bootstrap relations, minimal form-factors, and boundary states, the author shows how to compute current and vertex-operator correlators with controlled accuracy, and then translates these correlators into observable quantities such as AC conductance in fractional quantum Hall edge states and spin dynamics in the anisotropic Kondo model. The approach reveals rapid convergence of the form-factor series, clarifies UV/IR behavior, and provides practical, quantitatively accurate predictions across a range of strongly correlated impurity problems, including Friedel oscillations and Shiba-type relations. While powerful, the method faces technical hurdles near isotropic limits, yet remains broadly applicable to a variety of impurity-related phenomena in integrable field theories.

Abstract

These are notes of lectures given at The NATO Advanced Study Institute/EC Summer School on ``New Theoretical Approaches to Strongly Correlated Systems'' (Newton Institute, April 2000). They are a sequel to the notes I wrote two years ago for the Summer School ``Topological Aspects of Low Dimensional Systems'', (Les Houches, July 1998). In this second part, I review the form-factors technique and its extension to massless quantum field theories. I then discuss the calculation of correlators in integrable quantum impurity problems, with special emphasis on point contact tunneling in the fractional quantum Hall effect, and the two-state problem of dissipative quantum mechanics.

Paper Structure

This paper contains 14 sections, 156 equations, 6 figures.

Table of Contents

  1. Some generalities on form-factors
  2. Example: The sinh-Gordon model.
  3. Massless form-factors and the bulk current-current correlators.
  4. Current current correlators with a boundary
  5. The sine-Gordon Model.
  6. Conductance in the fractional quantum Hall effect
  7. General remarks.
  8. The free case.
  9. $G(\omega)$ at $g=1/3$.
  10. Anisotropic Kondo model and dissipative quantum mechanics.
  11. A computation of $C(t)$.
  12. Shiba's Relation.
  13. Friedel oscillations: correlations involving Vertex operators
  14. Conclusion

Figures (6)

  • Figure 1: One particle contribution.
  • Figure 2: Three particles contribution for $B=1,0.1$.
  • Figure 3: Frequency dependent conductance at T=0.
  • Figure 4: Spectral function for $T_B=0.1$.
  • Figure 5: Accuracy of the finite $T_B$ over the IR part of the envelope of $\rho(x)$ for $g=1/2$.
  • ...and 1 more figures