Ground-state energy and excitation spectrum of the Lieb-Liniger model : accurate analytical results and conjectures about the exact solution

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 \cite{Ristivojevic} 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

Tan's contact of a harmonically trapped one-dimensional Bose gas: strong-coupling expansion and conjectural approach at arbitrary interactions

We study Tan's contact, i.e. the coefficient of the high-momentum tails of the momentum distribution at leading order, for an interacting one-dimensional Bose gas subjected to a harmonic confinement. Using a strong-coupling systematic expansion of the ground-state energy of the homogeneous system stemming from the Bethe-Ansatz solution, together with the local-density approximation, we obtain the strong-coupling expansion for Tan's contact of the harmonically trapped gas. Also, we use a very accurate conjecture for the ground-state energy of the homogeneous system to obtain an approximate expression for Tan's contact for arbitrary interaction strength, thus estimating the accuracy of the strong-coupling expansion. Our results are relevant for ongoing experiments with ultracold atomic gases

Dynamic structure factor and drag force in a one-dimensional strongly-interacting Bose gas at finite temperature

We study the effect of thermal and quantum fluctuations on the dynamical response of a one-dimensional strongly-interacting Bose gas in a tight atomic waveguide. We combine the Luttinger liquid theory at arbitrary interactions and the exact Bose-Fermi mapping in the Tonks-Girardeau-impenetrable-boson limit to obtain the dynamic structure factor of the strongly-interacting fluid at finite temperature. Then, we determine the drag force felt by a potential barrier moving along the fluid in the experimentally realistic situation of finite barrier width and temperature.Comment: 13 pages, 11 figure

A connection between non-local one-body and local three-body correlations of the Lieb-Liniger model

We derive a connection between the fourth coefficient of the short-distance Taylor expansion of the one-body correlation function, and the local three-body correlation function of the Lieb-Liniger model of $\delta$-interacting spinless bosons in one dimension. This connection, valid at arbitrary interaction strength, involves the fourth moment of the density of quasi-momenta. Generalizing recent conjectures, we propose approximate analytical expressions for the fourth coefficient covering the whole range of repulsive interactions, validated by comparison with accurate numerics. In particular, we find that the fourth coefficient changes sign at interaction strength $\gamma_c\simeq 3.816$, while the first three coefficients of the Taylor expansion of the one-body correlation function retain the same sign throughout the whole range of interaction strengths

Integrability of natural Hamiltonian systems with homogeneous potentials of degree zero

We derive necessary conditions for integrability in the Liouville sense of natural Hamiltonian systems with homogeneous potential of degree zero. We derive these conditions through an analysis of the differential Galois group of variational equations along a particular solution generated by a non-zero solution \vd\in\C^n of nonlinear equations \grad V(\vd)=\vd. We proved that if the system integrable then the Hessian matrix V''(\vd) has only integer eigenvalues and is semi-simple.Comment: 13 page

Les ressources déployées par des leaders du développement durable : Quelles leçons pour l’éducation relative à l’environnement ?

Aujourd’hui, lors d’ingénieux projets développement durable, des leaders planifient et implantent des pratiques et des lieux bénéfiques à l’environnement. Ces initiatives représentent une source féconde d’informations au sujet des compétences d’aménagement du milieu qu’il serait pertinent de développer chez nos élèves. L’étude, se situant dans la perspective de poursuivre la compréhension des compétences à développer en éducation relative à l’environnement, visait l’identification des ressources liées à l’aménagement écologique du milieu, déployées par des leaders participant à cinq projets de développement durable (en urbanisme et en foresterie durable). L’analyse d’un document et des données d’entrevues individuelles d’explicitation montre que les principales ressources (cognitives, sociales et affectives) déployées par les leaders interrogés, relèvent non seulement de la résolution de problèmes, mais également d’une planification créative, complexe, souple, longitudinale, expérientielle, adaptée, patiente, collaborative, humaniste et environnementale. L’étude, dont on présente les retombées en ERE, montre enfin que les leaders déploient des ressources associées aux individus créatifs.In our day, leaders involved in ingenious sustainable development projects plan spaces and implement practices that are beneficial to the environment. These initiatives represent a fertile source of information on the competences linked to environmental design that we should nurture in our students. In view of improving our understanding of the competences that should be developed in environmental education (EE), this study sought to identify the competences used by leaders who participated in five sustainable development projects (in sustainable urban planning and forestry). The analysis of documents and data from individual interviews revealed that the main competences (cognitive, social and affective) used by the leaders interviewed, were not only related to problem solving but also to creative, complex, flexible, longitudinal, experiential, adapted, patient, collaborative, humanistic and environmental ways of planning. Finally, this study, of which we present the effects on EE, shows that leaders use competences which are associated with creative individuals

Reducing dynamic disorder in small-molecule organic semiconductors by suppressing large-amplitude thermal motions.

Thermal vibrations and the dynamic disorder they create can detrimentally affect the transport properties of van der Waals bonded molecular semiconductors. The low-energy nature of these vibrations makes it difficult to access them experimentally, which is why we still lack clear molecular design rules to control and reduce dynamic disorder. In this study we discuss the promising organic semiconductors rubrene, 2,7-dioctyl[1]benzothieno[3,2-b][1]benzothio-phene and 2,9-di-decyl-dinaphtho-[2,3-b:20,30-f]-thieno-[3,2-b]-thiophene in terms of an exceptionally low degree of dynamic disorder. In particular, we analyse diffuse scattering in transmission electron microscopy, to show that small molecules that have their side chains attached along the long axis of their conjugated core are better encapsulated in their crystal structure, which helps reduce large-amplitude thermal motions. Our work provides a general strategy for the design of new classes of very high mobility organic semiconductors with a low degree of dynamic disorder.S.I. acknowledges funding from the EPSRC, the Winton Programme for the Physics of Sustainability and the Cambridge Home and EU scholarship scheme (CHESS). G. S. acknowledges postdoctoral fellowship support from the Wiener-Anspach Foundation. We acknowledge the support of Nippon Kayaku in providing the materials C8-BTBT and C10-DNTT. We acknowledge Dr John Morrison for synthesis of TMTES-P and Marie Beatrice for her work that resulted in the thin-film structure of TMTES-P. We acknowledge Audrey Richard and Christian Ruzié for the synthesis of ditBu-BTBT and diTMS-BTBT.This is the final version of the article. It first appeared from Nature Publishing Group via https://doi.org/10.1038/ncomms1073