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arXiv:2209.15221 · National Science Review

Exact spectral function of one-dimensional Bose gases

Song Cheng, Yang-Yang Chen, Xi-Wen Guan, Wen‐Li Yang, Rubem Mondaini, Hai-Qing Lin

Published Jul 21, 2025• 4 citations
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Abstract

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.

DOI
10.1093/nsr/nwaf294

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•Jun 9, 2014•0 citations
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.
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.
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).
For the one-dimensional repulsive Bose gas (Lieb-Liniger model), we study a special class of highly-excited states obtained by giving a finite momentum to subgroups of particles. These states, which correspond to `splitting' the ground state Fermi sea-like quantum number configuration, are zero-entropy states which display interesting properties more normally associated to ground states. Using a numerically exact method based on integrability, we study these states' excitation spectrum, density correlations and momentum distribution functions. These correlations display power-law asymptotics, and are shown to be accurately described by an effective multicomponent Tomonaga-Luttinger liquid theory whose parameters are obtained from Bethe Ansatz. The non-universal correlation prefactors are moreover obtained from integrability, yielding a completely parameter-free fit of the correlator asymptotics.
The one-dimensional Lieb-Liniger Bose gas is a prototypical many-body system featuring universal Tomonaga-Luttinger liquid (TLL) physics and free fermion quantum criticality. We analytically calculate finite temperature local pair correlations for the strong-coupling Bose gas at quantum criticality using the polylog function in the framework of the Yang-Yang thermodynamic equations. We show that the local pair correlation has the universal value ${g}^{(2)}(0)\ensuremath{\approx}2p/(n\ensuremath{\varepsilon})$ in the quantum critical regime, the TLL phase, and the quasiclassical region, where ppp is the pressure per unit length rescaled by the interaction energy $\ensuremath{\varepsilon}=\frac{{\ensuremath{\hbar}}^{2}}{2m}{c}^{2}$ with interaction strength ccc and linear density nnn. This suggests the possibility to test finite temperature local pair correlations for the TLL in the relativistic dispersion regime and to probe quantum criticality with the local correlations beyond the TLL phase. Furthermore, thermodynamic properties at high temperatures are obtained by both high temperature and virial expansion of the Yang-Yang thermodynamic equation.
Interactions are known to have dramatic effects on bosonic gases in one dimension (1D). Not only does the ground state transform from a condensate-like state to an effective Fermi sea, but new fundamental excitations, which do not have any higher-dimensional equivalents, are predicted to appear. In this work, we trace these elusive excitations via their effects on the dynamical structure factor of 1D strongly-interacting Bose gases at low temperature. An array of 1D Bose gases is obtained by loading a 87^{87}87Rb condensate in a 2D lattice potential. The dynamical structure factor of the system is probed by energy deposition through low-momentum Bragg excitations. The experimental signals are compared to recent theoretical predictions for the dynamical structure factor of the Lieb-Liniger model at T>0T > 0T>0. Our results demonstrate that the main contribution to the spectral widths stems from the dynamics of the interaction-induced excitations in the gas, which cannot be described by the Luttinger liquid theory.