[Paper Review] One-body dynamical correlation function of Lieb-Liniger model at finite temperature

Song Cheng, Yang-Yang Chen+3 more|arXiv (Cornell University)|Nov 1, 2022

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper presents a comprehensive study of the finite-temperature one-body dynamical correlation function (1BDCF) in the Lieb-Liniger model using the form factor approach based on algebraic Bethe ansatz and numerical methods. It reveals how thermal fluctuations and interaction strength compete to shape spectral distributions, line-shapes, and static correlations, showing excellent agreement with Tomonaga-Luttinger liquid (TLL) theory for the one-body density matrix and identifying a distinct hump in the negative energy plane due to particle-hole excitations.

ABSTRACT

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.

Motivation & Objective

• To investigate the finite-temperature one-body dynamical correlation function (1BDCF) in the integrable Lieb-Liniger model with arbitrary interaction strength.
• To understand the interplay between thermal fluctuations and dynamical interactions in shaping spectral properties and correlation functions.
• To validate the applicability of Tomonaga-Luttinger liquid (TLL) theory for static correlations at finite temperature.
• To analyze the emergence of spectral features such as humps in the negative energy plane due to particle-hole excitations.
• To assess finite-size effects and ensure numerical convergence in the form factor approach.

Proposed method

• Employing the algebraic Bethe ansatz (ABA) to construct exact eigenstates and form factors for the Lieb-Liniger model.
• Using the form factor approach to compute the 1BDCF via summation over matrix elements of the field operator between eigenstates.
• Applying numerical techniques to evaluate the spectral representation of 1BDCF at finite temperature, including momentum distribution and one-body density matrix.
• Mapping the spectral distribution across the complex energy plane, particularly identifying features in the negative energy region.
• Comparing results with Tomonaga-Luttinger liquid (TLL) predictions for static correlations, especially the one-body density matrix.
• Conducting finite-size scaling analysis to verify convergence and minimize finite-size effects in numerical results.

Experimental results

Research questions

• RQ1 How do thermal fluctuations and interaction strength jointly influence the spectral distribution and line-shape of the 1BDCF in the Lieb-Liniger model at finite temperature?
• RQ2 What is the origin of the hump observed in the negative energy region of the 1BDCF, and how does it relate to particle-hole excitations?
• RQ3 To what extent do the static correlation functions—momentum distribution and one-body density matrix—agree with Tomonaga-Luttinger liquid (TLL) theory at finite temperature?
• RQ4 How do the spectral features of the 1BDCF evolve with increasing temperature and interaction strength?
• RQ5 What is the impact of finite system size on the accuracy of the 1BDCF calculation, and at what system size is convergence achieved?

Key findings

• The momentum distribution shows a clear suppression of low-momentum occupation at finite temperature compared to zero temperature, with stronger suppression for weaker interactions due to enhanced thermal fluctuations.
• A distinct hump emerges in the negative energy region of the 1BDCF at finite temperature, originating from 'jump-inward' particle-hole excitations relative to the thermal equilibrium state.
• The main peak in the 1BDCF is reduced in weight with increasing temperature, and the spectral weight is compensated by the newly formed hump in the negative energy plane.
• The one-body density matrix computed via the form factor approach shows excellent agreement with the TLL prediction, particularly in the regime where the low-lying spectrum is linearized.
• Finite-size analysis confirms that system sizes of N = L = 60 and N = L = 80 yield nearly identical results, with N = L = 40 showing noticeable deviations, indicating convergence at N = L = 60.
• The study demonstrates that the form factor approach combined with ABA and numerical evaluation enables accurate access to non-perturbative dynamical correlation functions in 1D integrable quantum gases at finite temperature.