One-particle density matrix and momentum distribution function of one-dimensional anyon gases

TL;DR

This paper analyzes the one-particle density matrix and momentum distribution of one-dimensional impenetrable anyon gases, revealing their mathematical structure and crossover behavior between bosons and fermions.

Contribution

It provides a detailed numerical and analytical study of the Green functions, showing the connection to Toeplitz determinants, Fisher-Hartwig conjecture, and Painleve VI equations.

Findings

Density matrix is a Toeplitz determinant with Fisher-Hartwig asymptotics.

Momentum distribution exhibits a crossover from bosonic to fermionic behavior.

The large momentum tail always scales as k^{-4} due to delta interactions.

Abstract

We present a systematic study of the Green functions of a one-dimensional gas of impenetrable anyons. We show that the one-particle density matrix is the determinant of a Toeplitz matrix whose large N asymptotic is given by the Fisher-Hartwig conjecture. We provide a careful numerical analysis of this determinant for general values of the anyonic parameter, showing in full details the crossover between bosons and fermions and the reorganization of the singularities of the momentum distribution function. We show that the one-particle density matrix satisfies a Painleve VI differential equation, that is then used to derive the small distance and large momentum expansions. We find that the first non-vanishing term in this expansion is always k^{-4}, that is proved to be true for all couplings in the Lieb-Liniger anyonic gas and that can be traced back to the presence of a delta function interaction in the Hamiltonian.