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Universality in Quantum Hall Systems: Coset Construction of Incompressible States

J. Froehlich, B. Pedrini, C. Schweigert, J. Walcher

TL;DR

The paper builds a unified CCFT/TFT framework for incompressible QHFs, identifying how bulk Chern-Simons data map to edge CCFT data and imposing consistency, modular-covariance, and stability criteria to select viable theories. It introduces a constructive program based on simple currents and coset techniques to generate CCFTs describing QHFs with rational Hall conductivities, and shows explicit coset-based examples that reproduce known plateaux such as $\sigma_H = \frac{1}{2}\frac{e^2}{h}$, $\frac{1}{4}\frac{e^2}{h}$, $\frac{3}{5}\frac{e^2}{h}$, and $\frac{e^2}{h}$. The work connects quantum Hall lattices to CCFT data, provides methods to gauge symmetries via cosets, and presents Virasoro minimal-model and simple-current extension realizations (including Ising-related states) as concrete candidates for incompressible QHF descriptions. Overall, the framework offers a systematic route to classify and construct incompressible QHF states and to understand their bulk-edge correspondence through rational CCFTs and their associated TFTs.

Abstract

Incompressible Quantum Hall fluids (QHF's) can be described in the scaling limit by three-dimensional topological field theories. Thanks to the correspondence between three-dimensional topological field theories and two dimensional chiral conformal field theories (CCFT's), we propose to study QHF's from the point of view of CCFT's. We derive consistency conditions and stability criteria for those CCFT's that can be expected to describe a QHF. A general algorithm is presented which uses simple currents to construct interesting examples of such CCFT's. It generalizes the description of QHF's in terms of Quantum Hall lattices. Explicit examples, based on the coset construction, provide candidates for the description of Quantum Hall fluids with Hall conductivity s_H=1/2 e^2/h, 1/4 e^2/h, 3/5 e^2/h, e^2/h,...

Universality in Quantum Hall Systems: Coset Construction of Incompressible States

TL;DR

The paper builds a unified CCFT/TFT framework for incompressible QHFs, identifying how bulk Chern-Simons data map to edge CCFT data and imposing consistency, modular-covariance, and stability criteria to select viable theories. It introduces a constructive program based on simple currents and coset techniques to generate CCFTs describing QHFs with rational Hall conductivities, and shows explicit coset-based examples that reproduce known plateaux such as , , , and . The work connects quantum Hall lattices to CCFT data, provides methods to gauge symmetries via cosets, and presents Virasoro minimal-model and simple-current extension realizations (including Ising-related states) as concrete candidates for incompressible QHF descriptions. Overall, the framework offers a systematic route to classify and construct incompressible QHF states and to understand their bulk-edge correspondence through rational CCFTs and their associated TFTs.

Abstract

Incompressible Quantum Hall fluids (QHF's) can be described in the scaling limit by three-dimensional topological field theories. Thanks to the correspondence between three-dimensional topological field theories and two dimensional chiral conformal field theories (CCFT's), we propose to study QHF's from the point of view of CCFT's. We derive consistency conditions and stability criteria for those CCFT's that can be expected to describe a QHF. A general algorithm is presented which uses simple currents to construct interesting examples of such CCFT's. It generalizes the description of QHF's in terms of Quantum Hall lattices. Explicit examples, based on the coset construction, provide candidates for the description of Quantum Hall fluids with Hall conductivity s_H=1/2 e^2/h, 1/4 e^2/h, 3/5 e^2/h, e^2/h,...

Paper Structure

This paper contains 18 sections, 54 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. General features of Quantum Hall Fluids
  3. Electrodynamics of incompressible quantum Hall fluids
  4. Topological Field Theory and incompressible quantum Hall fluids
  5. Two-dimensional chiral conformal field theory and three-dimensional topological field theory
  6. Conditions on a chiral conformal field theory describing a Quantum Hall Fluid
  7. Consistency conditions
  8. Remarks on modular invariance and covariance
  9. Stability of the Incompressible Quantum Hall Fluid
  10. Quantum Hall Lattices
  11. Construction of chiral conformal theories describing incompressible Quantum Hall fluids
  12. Explicit construction with electrons as simple currents
  13. Coset and orbifold construction of Quantum Hall CCFT's
  14. Examples of Quantum Hall fluids with $\sigma_H=\frac{1}{2}\frac{e^2}{h},\frac{3}{5}\frac{e^2}{h}, \frac{e^2}{h},\ldots$
  15. Virasoro minimal models
  16. ...and 3 more sections

Figures (6)

  • Figure 1: A Feynman diagram or ribbon graph for some sources. One can recognize (1) a self-twist, (2) a double self-twist, (3) a pair production, (4) a pair annihilation, (5) two sources that fuse into one, and (6) a source that splits in two.
  • Figure 2: A ribbon graph for a state in ${current algebra H}_{\vec{z},\vec{\lambda}}[v_{\lambda}]$.
  • Figure 3: The ribbon graph $\bar{{current algebra G}}\amalg{current algebra G}'$ obtained from the ribbon graphs of ${current algebra G}$ and ${current algebra G}'$.
  • Figure 4: The link whose link invariant in the three sphere $S^3$ equals $S_{\lambda\mu}$.
  • Figure 5: Physically realized representations for ${current algebra C}=\mathrm{Vir}_1$; $e$ marks the electron representation, $m$ a multi-electron representation and $\bullet$ the representations that fulfill (\ref{['eq:recipemonodromy2']}) and hence appear in the theory.
  • ...and 1 more figures