Lectures on Non Perturbative Field Theory and Quantum Impurity Problems

TL;DR

The notes survey non-perturbative methods for 1+1D field theories applied to quantum impurity problems, emphasizing boundary conformal field theory and integrability. They develop a boundary sine-Gordon framework, analyze UV/IR fixed points, and show how massless scattering and the Thermodynamic Bethe Ansatz yield exact transport results for edge-state tunneling in the fractional quantum Hall effect. The work highlights exact, non-equilibrium transport predictions and the role of boundary entropy in impurity RG flows, culminating in explicit conductance formulas and comparisons with experiments. Together, these approaches provide a powerful, exact toolkit for understanding strongly correlated impurity physics in low dimensions with direct experimental relevance.

Abstract

These are lectures presented at the Les Houches Summer School ``Topology and Geometry in Physics'', July 1998. They provide a simple introduction to non perturbative methods of field theory in 1+1 dimensions, and their application to the study of strongly correlated condensed matter problems - in particular quantum impurity problems. The level is moderately advanced, and takes the student all the way to the most recent progress in the field: many exercises and additional references are provided. In the first part, I give a sketchy introduction to conformal field theory. I then explain how boundary conformal invariance can be used to classify and study low energy, strong coupling fixed points in quantum impurity problems. In the second part, I discuss quantum integrability from the point of view of perturbed conformal field theory, with a special emphasis on the recent ideas of massless scattering. I then explain how these ideas allow the computation of (experimentally measurable) transport properties in cross-over regimes. The case of edge states tunneling in the fractional quantum Hall effect is used throughout the lectures as an example of application.

Figures (17)

• Figure 1: We will think of quantum impurity problems in various ways. The figure on the left represents a $1+1$ quantum point of view where the bulk right and left degrees of freedom are confined to a half line, with the impurity at the origin. Altrnatively, because the theory is massless in the bulk, one can unfold this picture to get only right degrees of freedom on the full line, as indicated on the right.
• Figure 2: Alternatively again, one can go to imaginary time and obtain a $2$D statistical mechanics point of view, with a theory defined in a half plane, that often we will chose to be the upper complex plane. In this figure, the arrows are supposed to "represent" the bulk degrees of freedom; later, we will see that they can be associated with integrable quasiparticles.
• Figure 3: A schematic experimental set up to study point contact tunneling in the fractional quantum Hall effect (the magnetic field points towards the reader). Details are provided in the text.
• Figure 4: Some of the geometries used in the text.
• Figure 5: The contour manipulations that lead to the definition of a commutator in radial quantization.