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Modular Invariant Partition Functions in the Quantum Hall Effect

Andrea Cappelli, Guillermo R. Zemba

TL;DR

The authors construct modular-invariant partition functions for quantum Hall edge theories on an annulus, framing the problem in rational conformal field theory terms and exploiting the Witten–Chern–Simons correspondence. They show Laughlin edges are described by extended $U(1)$ algebras with fusion group $\mathbb{Z}_p$ and analyze both diagonal and non-diagonal invariants for hierarchical plateaus within $\widehat{U(1)}^{\otimes m}$ and $\widehat{U(1)}\otimes\widehat{SU(m)}_1$ theories. A key result is that non-diagonal invariants with extended symmetries can account for observed plateaus beyond Jain’s sequence, while minimal $W_{1+\infty}$ theories are not RCFTs and thus require alternative treatments. The work connects modular data to Wen’s topological order via Verlinde algebra and discusses impurity-independence of topological order in Abelian cases, offering a structured RCFT framework to interpret hierarchical quantum Hall states and potential non-Abelian edge theories. Overall, the paper provides a comprehensive RCFT-based classification of edge theories and demonstrates how modular invariance constrains, and in some cases expands, the phenomenology of quantum Hall plateaus.

Abstract

We study the partition function for the low-energy edge excitations of the incompressible electron fluid. On an annular geometry, these excitations have opposite chiralities on the two edges; thus, the partition function takes the standard form of rational conformal field theories. In particular, it is invariant under modular transformations of the toroidal geometry made by the angular variable and the compact Euclidean time. The Jain series of plateaus have been described by two types of edge theories: the minimal models of the W-infinity algebra of quantum area-preserving diffeomorphisms, and their non-minimal version, the theories with U(1)xSU(m) affine algebra. We find modular invariant partition functions for the latter models. Moreover, we relate the Wen topological order to the modular transformations and the Verlinde fusion algebra. We find new, non-diagonal modular invariants which describe edge theories with extended symmetry algebra; their Hall conductivities match the experimental values beyond the Jain series.

Modular Invariant Partition Functions in the Quantum Hall Effect

TL;DR

The authors construct modular-invariant partition functions for quantum Hall edge theories on an annulus, framing the problem in rational conformal field theory terms and exploiting the Witten–Chern–Simons correspondence. They show Laughlin edges are described by extended algebras with fusion group and analyze both diagonal and non-diagonal invariants for hierarchical plateaus within and theories. A key result is that non-diagonal invariants with extended symmetries can account for observed plateaus beyond Jain’s sequence, while minimal theories are not RCFTs and thus require alternative treatments. The work connects modular data to Wen’s topological order via Verlinde algebra and discusses impurity-independence of topological order in Abelian cases, offering a structured RCFT framework to interpret hierarchical quantum Hall states and potential non-Abelian edge theories. Overall, the paper provides a comprehensive RCFT-based classification of edge theories and demonstrates how modular invariance constrains, and in some cases expands, the phenomenology of quantum Hall plateaus.

Abstract

We study the partition function for the low-energy edge excitations of the incompressible electron fluid. On an annular geometry, these excitations have opposite chiralities on the two edges; thus, the partition function takes the standard form of rational conformal field theories. In particular, it is invariant under modular transformations of the toroidal geometry made by the angular variable and the compact Euclidean time. The Jain series of plateaus have been described by two types of edge theories: the minimal models of the W-infinity algebra of quantum area-preserving diffeomorphisms, and their non-minimal version, the theories with U(1)xSU(m) affine algebra. We find modular invariant partition functions for the latter models. Moreover, we relate the Wen topological order to the modular transformations and the Verlinde fusion algebra. We find new, non-diagonal modular invariants which describe edge theories with extended symmetry algebra; their Hall conductivities match the experimental values beyond the Jain series.

Paper Structure

This paper contains 16 sections, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Partition functions for the Laughlin plateaus
  3. Modular invariance conditions
  4. Modular invariance and the fusion algebra
  5. The topological order and its independence of impurities
  6. $W_{1+\infty}\ $ minimal and non-minimal theories
  7. The $c=2$ case
  8. Partition functions for hierarchical plateaus
  9. Diagonal modular invariant for $\widehat{U(1)}^{\otimes m}$ theories
  10. Diagonal invariant for the $\widehat{U(1)}\otimes\widehat{SU(m)}_1$ theories
  11. Minimal $W_{1+\infty}\ $ models
  12. Non-diagonal $\widehat{U(1)}\otimes\widehat{SU(m)}_1$ invariants
  13. Comparison with the experiments: higher order hierarchical pattern
  14. Conclusions
  15. Modular forms and functions
  16. ...and 1 more sections

Figures (1)

  • Figure 1: Experimentally observed plateaus: their Hall conductivity $\sigma_H=(e^2/h)\nu$ is displayed in units of $(e^2/h)$. The marks $\ (\bullet)\ $ denote stable (i.e. large) plateaus, which have been seen in several experiments; the marks $\ (\circ)\ $ denote less developed plateaus and plateaus found in one experiment only. Plateaus belonging to the Jain main series have their filling fraction in bold type, the other ones are in italics. Coexisting fluids at the same filling fraction have been found at $\nu=2/3,2/5,3/5,5/7$.