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arXiv:1403.3415 · Physical Review Letters

Excitation Spectrum of the Lieb-Liniger Model

Zoran Ristivojević

Published Jul 2, 2014• 60 citations
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Abstract

We study the integrable model of one-dimensional bosons with contact repulsion. In the limit of weak interaction, we use the microscopic hydrodynamic theory to obtain the excitation spectrum. The statistics of quasiparticles changes with the increase of momentum. At lowest momenta good quasiparticles are fermions, while at higher momenta they are Bogoliubov bosons, in accordance with recent studies. In the limit of strong interaction, we analyze the exact solution and find exact results for the spectrum in terms of the asymptotic series. Those results undoubtedly suggest that fermionic quasiparticle excitations actually exist at all momenta for moderate and strong interaction and also at lowest momenta for arbitrary interaction. Moreover, at strong interaction we find highly accurate analytical results for several relevant quantities of the Lieb-Liniger model.

DOI
10.1103/physrevlett.113.015301

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We study the integrable model of one-dimensional bosons with contact repulsion. In the limit of weak interaction, we use the microscopic hydrodynamic theory to obtain the excitation spectrum. The statistics of quasiparticles changes with the increase of momentum. At lowest momenta good quasiparticles are fermions, while at higher momenta they are Bogoliubov bosons, in accordance with recent studies. In the limit of strong interaction, we analyze the exact solution and find exact results for the spectrum in terms of the asymptotic series. Those results undoubtedly suggest that fermionic quasiparticle excitations actually exist at all momenta for moderate and strong interaction and also at lowest momenta for arbitrary interaction. Moreover, at strong interaction we find highly accurate analytical results for several relevant quantities of the Lieb-Liniger model.
We study the ground-state properties and excitation spectrum of the Lieb-Liniger model, i.e. the one-dimensional Bose gas with repulsive contact interactions. We solve the Bethe-Ansatz equations in the thermodynamic limit by using an analytic method based on a series expansion on orthogonal polynomials developed in and push the expansion to an unprecedented order. By a careful analysis of the mathematical structure of the series expansion, we make a conjecture for the analytic exact result at zero temperature and show that the partially resummed expressions thereby obtained compete with accurate numerical calculations. This allows us to evaluate the density of quasi-momenta, the ground-state energy, the local two-body correlation function and Tan’s contact. Then, we study the two branches of the excitation spectrum. Using a general analysis of their properties and symmetries, we obtain novel analytical expressions at arbitrary interaction strength which are found to be extremely accurate in a wide range of intermediate to strong interactions.
We study the decay of Bogoliubov quasiparticles in one-dimensional Bose gases. Starting from the hydrodynamic Hamiltonian, we develop a microscopic theory that enables one to systematically study both the excitations and their decay. At zero temperature, the leading mechanism of decay of a quasiparticle is disintegration into three others. We find that low-energy quasiparticles (phonons) decay with the rate that scales with the seventh power of momentum, whereas the rate of decay of the high-energy quasiparticles does not depend on momentum. In addition, our approach allows us to study analytically the quasiparticle decay in the whole crossover region between the two limiting cases. When applied to integrable models, including the Lieb-Liniger model of bosons with contact repulsion, our theory confirms the absence of the decay of quasiparticle excitations. We account for two types of integrability-breaking perturbations that enable finite decay: three-body interaction between the bosons and two-body interaction of finite range.
Abstract The Lieb–Liniger model describes one-dimensional bosons with contact interactions. This many-body system admits an exact solution in terms of the Bethe ansatz. Some of the exact and perturbative results for this model are reviewed. Particular attention is devoted to the explicit evaluation, in terms of the interaction parameter, of physical quantities that can be formally exactly extracted from the Bethe ansatz solution. Another goal of this review is to stress exact relations between various quantities. The technical developments are explained in detail. The most relevant experimental realisations of the studied problems are eventually discussed. This review also contains several new results such as the study of convergence of the ground-state energy series at strong interactions, the excitation spectrum at high energies, and the evaluation of the boundary energy.
In a one-dimensional weakly interacting Bose-Fermi mixture, one branch of elementary excitations is well described by the Bogoliubov spectrum. Here we use the microscopic theory to study the decay of such quasiparticle excitations. The main scattering process which leads to their decay is the backscattering of a Bogoliubov quasiparticle off the Fermi sea, where a particle-hole pair is excited. For a low-momentum quasiparticle (phonon) of momentum qqq, we find that the decay rate scales as q3{q}^{3}q3 provided qqq is smaller than the Fermi momentum kF{k}_{F}kF​, while in the opposite case the decay behaves as q2{q}^{2}q2. If the ratio of the masses of fermions and bosons is equal to the ratio of the boson-fermion and the boson-boson interaction strengths, the decay rate changes dramatically. It scales as q7{q}^{7}q7 for q<kFq<{k}_{F}q<kF​, while we find q6{q}^{6}q6 scaling at q>kFq>{k}_{F}q>kF​. For a high-momentum Bogoliubov quasiparticle, we find a constant decay rate for q<kFq<{k}_{F}q<kF​, while it scales as 1/q1/q1/q for q>kFq>{k}_{F}q>kF​. We also find an analytic expression for the decay rate in the crossover region between low and high momenta. The decay rate is a continuous, but nonanalytic function of the momentum at q=kFq={k}_{F}q=kF​. In the special case when the parameters of our system correspond to the integrable model, we observe that the decay rate vanishes.
We develop an alternative description to solve the problem of the ground-state energy of the Lieb-Liniger model that describes one-dimensional bosons with contact repulsion. For this integrable model we express the Lieb integral equation in the representation of Chebyshev polynomials. The latter form is convenient to efficiently obtain very precise numerical results in the singular limit of weak interaction. Such highly precise data enable us to use the integer relation algorithm to discover the analytical form of the coefficients in the expansion of the ground-state energy for small values of the interaction parameter. We obtained the first nine terms of the expansion using quite moderate numerical efforts. The detailed knowledge of behavior of the ground-state energy on the interaction immediately leads to exact perturbative results for the excitation spectrum.
The Lieb-Liniger model describes bosons with contact interactions in one-dimensional space. In the limit of weak repulsive particle interactions, there are two types of low-lying excitation spectrum. The first is reproduced by the Bogoliubov dispersion relation, and the other is believed to correspond to dark soliton excitations. While there are various evidences that the type II spectrum is related to dark solitons, it has not been shown that measurements of positions of particles reveal dark soliton density profiles. Here, we employ the Bethe ansatz approach and show that dark solitons emerge in the measurement process if the system is prepared in an eigenstate corresponding to the type II spectrum. We analyze single and double dark solitons as well as weak and strong interaction regimes.