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arXiv:0803.0750 · Journal of Physics A Mathematical and Theoretical

One-dimensional impenetrable anyons in thermal equilibrium: II. Determinant representation for the dynamic correlation functions

Ovidiu I Pâţu, Vladimir E Korepin, D. V. Averin

Published May 28, 2008• 54 citations
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Abstract

We have obtained a determinant representation for the time- and temperature-dependent field–field correlation function of the impenetrable Lieb–Liniger gas of anyons through direct summation of the form factors. In the static case, the obtained results are shown to be equivalent to those that follow from the anyonic generalization of Lenard's formula.

DOI
10.1088/1751-8113/41/25/255205

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We have obtained a determinant representation for the time- and temperature-dependent field–field correlation function of the impenetrable Lieb–Liniger gas of anyons through direct summation of the form factors. In the static case, the obtained results are shown to be equivalent to those that follow from the anyonic generalization of Lenard's formula.
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 .
Using the determinant representation for the field-field correlation functions of impenetrable anyons at finite temperature obtained in a previous paper, we derive a system of nonlinear partial differential equations completely characterizing the correlators. The system is the same as the one for impenetrable bosons but with different initial conditions. The large-distance asymptotic behavior of the correlation functions is obtained from the analysis of the Riemann-Hilbert problem associated with the system of differential equations. We calculate both the exponential and pre-exponential factors in the asymptotics of the field-field correlators. The asymptotics derived in this way agree with those of the free fermions and impenetrable bosons in the appropriate limits, κ→1κ\to 1κ→1 and κ→0κ\to 0κ→0, of the statistics parameter κκκ, and coincide with the predictions of the conformal field theory at low temperatures.
We study thermal correlation functions of the one-dimensional impenetrable lattice anyons. These correlation functions can be presented as a difference of two Fredholm determinants. To describe the large-time and long-distance behavior of these objects, we use the effective form-factor approach. The asymptotic behavior is different in the spacelike and timelike regions. In particular, in the timelike region we observe the additional power factor on top of the exponential decay. We argue that this result is universal as it is related to the discontinuous behavior of the phase shift function of the effective fermions. At particular values of the anyonic parameter, we recover the asymptotics of the spin-spin correlation functions in the XXXXXX quantum chain.
The momentum- and frequency-dependent one-body correlation function of the one-dimensional interacting Bose gas (Lieb–Liniger model) in the repulsive regime is studied using the Algebraic Bethe Ansatz and numerics. We first provide a determinant representation for the field form factor which is well adapted to numerical evaluation. The correlation function is then reconstructed to high accuracy for systems with finite but large numbers of particles, for a wide range of values of the interaction parameter. Our results are extensively discussed, in particular their specialization to the static case.
Abstract. The momentum- and frequency-dependent one-body correlation function of the one-dimensional interacting Bose gas (Lieb-Liniger model) in the repulsive regime is studied using the Algebraic Bethe Ansatz and numerics. We first provide a determinant representation for the field form factor which is welladapted to numerical evaluation. The correlation function is then reconstructed to high accuracy for systems with finite but large numbers of particles, for a wide range of values of the interaction parameter. Our results are extensively discussed, in particular their specialization to the static case. One-particle dynamical correlations in the one-dimensional Bose gas 2 1.
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.