One-Dimensional Impenetrable Anyons in Thermal Equilibrium. I. Anyonic Generalization of Lenard's Formula

TL;DR

This paper generalizes Lenard's formula to one-dimensional impenetrable anyons, expressing their correlation functions in terms of fermionic matrices, revealing unique parity-dependent properties due to anyonic statistics.

Contribution

It extends Lenard's fermionic correlation framework to anyons, providing explicit formulas for their reduced density matrices in thermal equilibrium.

Findings

Derived Fredholm minor expressions for anyonic density matrices.

Revealed parity-dependent differences in anyonic field correlators.

Unified bosonic and anyonic cases within a common mathematical framework.

Abstract

We have obtained an expansion of the reduced density matrices (or, equivalently, correlation functions of the fields) of impenetrable one-dimensional anyons in terms of the reduced density matrices of fermions using the mapping between anyon and fermion wavefunctions. This is the generalization to anyonic statistics of the result obtained by A. Lenard for bosons. In the case of impenetrable but otherwise free anyons with statistical parameter $\kappa$, the anyonic reduced density matrices in the grand canonical ensemble is expressed as Fredholm minors of the integral operator ($1-\gamma \hat \theta_T$) with complex statistics-dependent coefficient $\gamma=(1+e^{\pm i\pi\kappa})/ \pi$. For $\kappa=0$ we recover the bosonic case of Lenard $\gamma=2/\pi$. Due to nonconservation of parity, the anyonic field correlators $\la \fad(x')\fa(x)\ra$ are different depending on the sign of $x'-x$.