Exact results of dynamical structure factor of Lieb-Liniger model

TL;DR

This paper precisely calculates the dynamical structure factor of the Lieb-Liniger model for large particle numbers using algebraic Bethe ansatz, revealing spectral features and threshold behaviors consistent with nonlinear TLL theory.

Contribution

It introduces a form factor approach based on algebraic Bethe ansatz to compute the DSF for large systems with arbitrary interaction strength, surpassing previous limitations.

Findings

Accurate DSF line-shape for systems with up to 2000 particles.

Emergence of power-law singularities near spectral thresholds.

Validation of nonlinear TLL theory through threshold behavior analysis.

Abstract

The dynamical structure factor (DSF) represents a measure of dynamical density-density correlations in a quantum many-body system. Due to the complexity of many-body correlations and quantum fluctuations in a system of an infinitely large Hilbert space, such kind of dynamical correlations often impose a big theoretical challenge. For one dimensional (1D) quantum many-body systems, qualitative predictions of dynamical response functions are usually carried out by using the Tomonaga-Luttinger liquid (TLL) theory. In this scenario, a precise evaluation of the DSF for a 1D quantum system with arbitrary interaction strength remains a formidable task. In this paper, we use the form factor approach based on algebraic Bethe ansatz theory to calculate precisely the DSF of Lieb-Liniger model with an arbitrary interaction strength at a large scale of particle number. We find that the DSF for a system as large as 2000 particles enables us to depict precisely its line-shape from which the power-law singularity with corresponding exponents in the vicinities of spectral thresholds naturally emerge. It should be noted that, the advantage of our algorithm promises an access to the threshold behavior of dynamical correlation functions, further confirming the validity of nonlinear TLL theory besides Kitanine et. al. 2012 J. Stat. Mech. P09001. Finally we discuss a comparison of results with the results from the ABACUS method by J.-S. Caux 2009 J. Math. Phys. 50 095214 as well as from the strongly coupling expansion by Brand and Cherny 2005 Phys. Rev. A 72 033619.