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arXiv:1010.2268 · Physical Review B

Exact prefactors in static and dynamic correlation functions of one-dimensional quantum integrable models: Applications to the Calogero-Sutherland, Lieb-Liniger, and X X Z models

Aditya Shashi, Miłosz Panfil, Jean-Sébastien Caux, Adilet Imambekov

Published Apr 24, 2012• 77 citations
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Abstract

In this paper, we demonstrate a recently developed technique which addresses the problem of obtaining nonuniversal prefactors of the correlation functions of one-dimensional (1D) systems at zero temperature. Our approach combines the effective field theory description of generic 1D quantum liquids with the finite-size scaling of form factors (matrix elements) which are obtained using microscopic techniques developed in the context of integrable models. We thus establish exact analytic forms for the prefactors of the long-distance behavior of equal-time correlation functions as well as prefactors of singularities of dynamic response functions. In this paper, our focus is on three specific integrable models: the Calogero-Sutherland, Lieb-Liniger, and $XXZ$ models.

DOI
10.1103/physrevb.85.155136

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We develop a general approach to calculating ``nonuniversal'' prefactors in static and dynamic correlation functions of one-dimensional (1D) quantum liquids at zero temperature by relating them to the finite-size scaling of certain matrix elements (form factors). This represents a powerful tool for extracting data valid in the thermodynamic limit from finite-size effects. As the main application, we consider weakly interacting spinless fermions with an arbitrary pair interaction potential, for which we perturbatively calculate certain prefactors in static and dynamic correlation functions. We also evaluate prefactors of the long-distance behavior of correlation functions nonperturbatively for the exactly solvable Lieb-Liniger model of 1D bosons.
The dynamical correlated properties of one-dimensional (1D) Bose gases provide profound understanding of novel physics emergent from collective excitations, for instance, the breakdown of off-diagonal long-range order, and the establishment of Tomonaga-Luttinger liquid theory. However, due to the nonperturbative nature of 1D many-body systems, the exact evaluation of correlation functions is notoriously difficult. Here, by means of a form factor approach based on an algebraic Bethe ansatz and numerics, we present a thorough study on the one-body dynamical correlation function (1BDCF) of the Lieb-Liniger model at finite temperature. The influence of thermal fluctuation and interaction on the behavior of 1BDCF has been demonstrated and analyzed from various perspectives, including the spectral distribution, the line shape at fixed momentum, and the corresponding static correlations.
The physics of the attractive one-dimensional Bose gas (Lieb–Liniger model) is investigated with techniques based on the integrability of the system. Combining a knowledge of particle quasi-momenta to exponential precision in the system size with determinant representations of matrix elements of local operators coming from the algebraic Bethe ansatz, we obtain rather general analytical results for the zero-temperature dynamical correlation functions of the density and field operators. Our results thus provide quantitative predictions for possible future experiments in atomic gases or optical waveguides.
Abstract. The physics of the attractive one-dimensional Bose gas (Lieb-Liniger model) is investigated with techniques based on the integrability of the system. Combining a knowledge of particle quasi-momenta to exponential precision in the system size with determinant representations of matrix elements of local operators coming from the Algebraic Bethe Ansatz, we obtain rather general analytical results for the zero-temperature dynamical correlation functions of the density and field operators. Our results thus provide quantitative predictions for possible future experiments in atomic gases or optical waveguides. Dynamics of the attractive 1D Bose gas: analytical treatment from integrability 2
We develop a form factor approach to the study of dynamical correlation functions of quantum integrable models in the critical regime. As an example, we consider the quantum non-linear Schrödinger model. We derive the long-distance/long-time asymptotic behavior of various two-point functions of this model. We also compute edge exponents and amplitudes characterizing the power-law behavior of dynamical response functions on the particle–hole excitation thresholds. These last results confirm predictions based on the non-linear Luttinger liquid method. Our results rely on a first principles derivation, based on a microscopic analysis of the model, without invoking, at any stage, any correspondence with a continuous field theory. Furthermore, our approach only makes use of certain general properties of the model, so that it should be applicable, possibly with minor modifications, to a wide class of (not necessarily integrable) gapless one-dimensional Hamiltonians.
Exactly solved models provide rigorous understanding of many-body phenomena in strongly correlated systems. In this article, we report a breakthrough in uncovering universal many-body correlated properties of the quantum integrable Lieb-Liniger model. We exactly calculate the dynamical correlation functions by computing the form factors through a newly developed method, by which we are capable of calculating all possible 'relative excitations' over the ground state or a finite temperature state to high precision. Consequently, full spectral functions obtained for the model manifest the unique power-law singularity behaviour at the spectral threshold, confirming the validity of nonlinear Luttinger liquid theory. Our method advances the theory of dynamical correlation functions with high precision towards the thermodynamic limit, and is capable of benchmarking experimental observation of such novel correlated properties.