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arXiv:0812.3264 · Physical Review A

Dynamic and static density-density correlations in the one-dimensional Bose gas: Exact results and approximations

Alexander Yu. Cherny, Joachim Brand

Published Apr 10, 2009• 34 citations
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Abstract

We discuss approximate formulas for the dynamic structure factor of the one-dimensional Bose gas in the Lieb-Liniger model that appear to be applicable over a wide range of the relevant parameters such as the interaction strength, frequency, and wave number. The suggested approximations are consistent with the exact results known in limiting cases. In particular, we encompass exact edge exponents as well as the Luttinger-liquid and perturbation theoretic results. We further discuss derived approximations for the static structure factor and the pair distribution function $g(x)$. The approximate expressions show excellent agreement with numerical results based on the algebraic Bethe ansatz.

DOI
10.1103/physreva.79.043607

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We discuss approximate formulas for the dynamic structure factor of the one-dimensional Bose gas in the Lieb-Liniger model that appear to be applicable over a wide range of the relevant parameters such as the interaction strength, frequency, and wave number. The suggested approximations are consistent with the exact results known in limiting cases. In particular, we encompass exact edge exponents as well as the Luttinger-liquid and perturbation theoretic results. We further discuss derived approximations for the static structure factor and the pair distribution function g(x)g(x)g(x). The approximate expressions show excellent agreement with numerical results based on the algebraic Bethe ansatz.
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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.
Recently, Imambekov and Glazman [Phys. Rev. Lett. 100, 206805 (2008)] showed that the dynamic structure factor (DSF) of the 1D Bose gas demonstrates power-law behaviour along the limiting dispersion curve of the collective modes and calculated the corresponding exponents exactly. Combining these recent results with a previously obtained strong-coupling expansion we present an interpolation formula for the DSF of the 1D Bose gas. The obtained expression is further consistent with exact low energy exponents from Luttinger liquid theory and shows nice agreement with recent numerical results.
Correlation functions related to the dynamic density response of the one-dimensional Bose gas in the model of Lieb and Liniger are calculated. An exact Bose-Fermi mapping is used to work in a fermionic representation with a pseudopotential Hamiltonian. The Hartree-Fock and generalized random phase approximations are derived and the dynamic polarizability is calculated. The results are valid to first order in $1∕\ensuremath{\gamma}$, where $\ensuremath{\gamma}$ is Lieb-Liniger coupling parameter. Approximations for the dynamic and static structure factor at finite temperature are presented. The results preclude superfluidity at any finite temperature in the large-$\ensuremath{\gamma}$ regime due to the Landau criterion. Due to the exact Bose-Fermi duality, the results apply for spinless fermions with weak ppp-wave interactions as well as for strongly interacting bosons.