[Paper Review] The universal chiral partition function for exclusion statistics

Alexander Bérkovich, Barry M. McCoy|ArXiv.org|Aug 4, 1998

Advanced NMR Techniques and Applications53 references16 citations

TL;DR

This paper establishes a direct equivalence between Haldane's exclusion statistics and the universal chiral partition function derived from conformal field theory and Rogers-Ramanujan identities. By showing that the fermionic counting rules (7)–(9) generalize Haldane's linear exclusion rule (1), the authors propose that these counting rules—previously used in CFT and lattice models—constitute a more natural and universal definition of exclusion statistics in one dimension.

ABSTRACT

We demonstrate the equality between the universal chiral partition function, which was first found in the context of conformal field theory and Rogers-Ramanujan identities, and the exclusion statistics introduced by Haldane in the study of the fractional quantum Hall effect. The phenomena of multiple representations of the same conformal field theory by different sets of exclusion statistics is discussed in the context of the ${\hat u}(1)$ theory of a compactified boson of radius $R.$

Motivation & Objective

• To establish a rigorous connection between Haldane's exclusion statistics and the universal chiral partition function in conformal field theory.
• To resolve the long-standing ambiguity in defining exclusion statistics by showing that fermionic counting rules (7)–(9) generalize Haldane's rule (1).
• To demonstrate that the universal chiral partition function (2) with u=∞ and y=1 corresponds to the exclusion statistics of anyons and fractional statistics in 1D systems.
• To argue that the fermionic counting rules (7)–(9) are a more fundamental and general definition of exclusion statistics than Haldane's original formulation (1).
• To clarify the role of different regularization schemes in constructing massless theories and their impact on particle statistics in the fractional quantum Hall effect.

Proposed method

• Derives the universal chiral partition function (2) as a grand partition function for n species of chiral particles with fugacities y_α, using Gaussian polynomials and q-series identities.
• Applies fermionic counting rules (7)–(9) to define the allowed momenta for particles with Pauli exclusion, ensuring no two particles occupy the same state.
• Uses the limiting form of Gaussian polynomials (4) to recover the standard fermionic and bosonic partition functions (11) and (12) in the q→0 limit.
• Demonstrates that the universal chiral partition function (2) with u=∞ reduces to the exclusion statistics of Haldane (1) when the matrix B encodes the statistical interaction g_αβ.
• Analyzes the modular properties and dilogarithm identities (31) to show that only specific values of B (e.g., b=2,1,1/2) correspond to known conformal field theories like M(2,5), M(3,4), M(3,5).
• Compares different fermionic descriptions of the same CFT (e.g., M(p,p+1)) to show that they arise from different ultraviolet regularizations and correspond to different exclusion statistics.

Experimental results

Research questions

• RQ1 How can Haldane's exclusion statistics be mapped to the universal chiral partition function used in conformal field theory?
• RQ2 What is the precise mathematical relationship between the fermionic counting rules (7)–(9) and Haldane's linear exclusion rule (1)?
• RQ3 Why have the connections between exclusion statistics and Rogers-Ramanujan identities not been widely recognized in the fractional quantum Hall effect literature?
• RQ4 In what sense is the universal chiral partition function truly 'universal' beyond its original CFT context?
• RQ5 Can the fermionic counting rules (7)–(9) be considered a more fundamental definition of exclusion statistics than Haldane's original formulation?

Key findings

• The universal chiral partition function (2) with u=∞ and y=1 is mathematically equivalent to Haldane's exclusion statistics, with the matrix B encoding the statistical interaction g_αβ.
• The fermionic counting rules (7)–(9) generalize Haldane's rule (1), and the authors propose that these rules should replace (1) as the standard definition of exclusion statistics in one dimension.
• For n=1, only three values of B (b=2,1,1/2) correspond to known conformal field theories: M(2,5), M(3,4), and M(3,5), respectively.
• The dilogarithm identities (31) in the conformal limit are rational multiples of L(1), a property that holds only for these three special cases.
• The partition function (2) transforms under a modular group representation only for these three special cases, confirming their physical significance in Kac–Moody and affine Lie algebra theories.
• Different fermionic descriptions of the same CFT (e.g., M(p,p+1)) arise from different ultraviolet regularization schemes, leading to distinct exclusion statistics that are non-local with respect to one another.