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arXiv:cond-mat/0703117 · Physical Review B

Correlation functions of one-dimensional anyonic fluids

Pasquale Calabrese, Mihail Mintchev

Published Jun 29, 2007• 81 citations
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Abstract

A universal description of correlation functions of one-dimensional anyonic gapless systems in the low-momentum regime is presented. We point out a number of interesting features, including universal oscillating terms with frequency proportional to the statistical parameter and beating effects close to the fermion points. The results are applied to the one-dimensional anyonic Lieb-Liniger model and checked against the exact results in the impenetrable limit.

DOI
10.1103/physrevb.75.233104

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A universal description of correlation functions of one-dimensional anyonic gapless systems in the low-momentum regime is presented. We point out a number of interesting features, including universal oscillating terms with frequency proportional to the statistical parameter and beating effects close to the fermion points. The results are applied to the one-dimensional anyonic Lieb-Liniger model and checked against the exact results in the impenetrable limit.
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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 Lenard (1966 J. Math. Phys. 7 1268) for bosons. In the case of impenetrable but otherwise free anyons with statistics parameter κ, the anyonic reduced density matrices in the grand canonical ensemble are expressed as Fredholm minors of the integral operator with the complex statistics-dependent coefficient γ = (1 + e ±iπκ )/π. For κ = 0 we recover the bosonic case of Lenard γ = 2/π. Due to nonconservation of parity, the anyonic field correlators ⟨Ψ† A ( x ')Ψ A ( x )⟩ are different depending on the sign of x ' − x .
We have investigated the properties of a model of 1D anyons interacting through a δ-function repulsive potential. The structure of the quasi-periodic boundary conditions for the anyonic field operators and the many-anyon wavefunctions is clarified. The spectrum of the low-lying excitations including the particle–hole excitations is calculated for periodic and twisted boundary conditions. Using the ideas of the conformal field theory we obtain the large-distance asymptotics of the density and field correlation function at the critical temperature T = 0 and at small finite temperatures. Our expression for the field correlation function extends the results in the literature obtained for harmonic quantum anyonic fluids.
This work presents the derivation of the large time and distance asymptotic behavior of the field-field correlation functions of impenetrable one-dimensional anyons at finite temperature. In the appropriate limits of the statistics parameter, we recover the well-known results for impenetrable bosons and free fermions. In the low-temperature (usually expected to be the "conformal") limit, and for all values of the statistics parameter away from the bosonic point, the leading term in the correlator does not agree with the prediction of the conformal field theory, and is determined by the singularity of the density of the single-particle states at the bottom of the single-particle energy spectrum.