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Fractional index of Bargmann-Fock space and Landau levels

Guo Chuan Thiang

TL;DR

This work analyzes the Bargmann-Fock (lowest Landau level) space to expose exact integer quantization of Hall conductance via CHHP index theory and then uncovers a hidden rational structure in higher-order commutator traces. By computing the principal function for the Fock–Toeplitz pair and establishing a unit-square support, it shows that traces of polynomial commutators are rational and that higher Landau levels inherit additivity. The results connect direct, elementary calculations with noncommutative index theory, extracting exact numerical quantization and suggesting a link to fractional quantization through fractional traces and anyonic perspectives. The findings offer a principled framework for understanding both integer and potential fractional Hall effects in magnetic systems, with implications for lattice models and coarse-index methods in topological transport.

Abstract

The lowest Landau level Hilbert space, or the Bargmann-Fock space, admits a quantized trace for the commutator of its position coordinate operators. We exploit the Carey-Pincus theory of principal functions of trace class commutators to probe this integer quantization result further, and uncover a hidden rational structure in the higher-order commutator-traces. This shows how exact fractional quantization can occur whenever exact integral quantization does.

Fractional index of Bargmann-Fock space and Landau levels

TL;DR

This work analyzes the Bargmann-Fock (lowest Landau level) space to expose exact integer quantization of Hall conductance via CHHP index theory and then uncovers a hidden rational structure in higher-order commutator traces. By computing the principal function for the Fock–Toeplitz pair and establishing a unit-square support, it shows that traces of polynomial commutators are rational and that higher Landau levels inherit additivity. The results connect direct, elementary calculations with noncommutative index theory, extracting exact numerical quantization and suggesting a link to fractional quantization through fractional traces and anyonic perspectives. The findings offer a principled framework for understanding both integer and potential fractional Hall effects in magnetic systems, with implications for lattice models and coarse-index methods in topological transport.

Abstract

The lowest Landau level Hilbert space, or the Bargmann-Fock space, admits a quantized trace for the commutator of its position coordinate operators. We exploit the Carey-Pincus theory of principal functions of trace class commutators to probe this integer quantization result further, and uncover a hidden rational structure in the higher-order commutator-traces. This shows how exact fractional quantization can occur whenever exact integral quantization does.
Paper Structure (15 sections, 3 theorems, 49 equations, 1 figure)

This paper contains 15 sections, 3 theorems, 49 equations, 1 figure.

Table of Contents

  1. Overview
  2. Trace of commutator of Toeplitz operators on Fock space
  3. Bargmann--Fock space
  4. Fock--Toeplitz operators and their commutator-traces
  5. Index-theoretic connections and rational traces
  6. Carey--Helton--Howe--Pincus index theory
  7. Rational quantization of commutator-trace of Fock--Toeplitz operators
  8. Exact quantization of Hall conductance
  9. Hall conductance of Landau levels
  10. Stability and macroscopic nature of index
  11. Magnetic lattice fermions
  12. Higher Landau levels and additivity
  13. Exact numerical quantization
  14. Fractional traces
  15. Fractional conductance

Key Result

Proposition 2.2

Let $\Lambda_1,\Lambda_2:\mathbb{R}\to[0,1]$ be any switch functions, so are switch functions of the first and second coordinates, respectively. Then

Figures (1)

  • Figure 1: For a switch function $\Lambda$, the integral \ref{['eqn:switch.integral']} is the signed area between the graph of $\Lambda$ and the graph of its $a$-translate, which equals $a$.

Theorems & Definitions (11)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Example 3.1
  • Remark 3.2
  • Theorem 3.3
  • Corollary 3.4
  • proof
  • Remark 3.5
  • Remark 4.1
  • ...and 1 more