sl_2 Gaudin model with Jordanian twist

sl_2 Gaudin model with Jordanian twist is studied. This system can be obtained as the semiclassical limit of the XXX spin chain deformed by the Jordanian twist. The appropriate creation operators that yield the Bethe states of the Gaudin model and consequently its spectrum are defined. Their commutation relations with the generators of the corresponding loop algebra as well as with the generating function of integrals of motion are given. The inner products and norms of Bethe states and the relation to the solutions of the Knizhnik-Zamolodchikov equations are discussed.Comment: 22 pages; corrected typo

Invariants of Collective Neutrino Oscillations

We consider the flavor evolution of a dense neutrino gas by taking into account both vacuum oscillations and self interactions of neutrinos. We examine the system from a many-body perspective as well as from the point of view of an effective one-body description formulated in terms of the neutrino polarization vectors. We show that, in the single angle approximation, both the many-body picture and the effective one-particle picture possess several constants of motion. We write down these constants of motion explicitly in terms of the neutrino isospin operators for the many-body case and in terms of the polarization vectors for the effective one-body case. The existence of these constants of motion is a direct consequence of the fact that the collective neutrino oscillation Hamiltonian belongs to the class of Gaudin Hamiltonians. This class of Hamiltonians also includes the (reduced) BCS pairing Hamiltonian describing superconductivity. We point out the similarity between the collective neutrino oscillation Hamiltonian and the BCS pairing Hamiltonian. The constants of motion manifest the exact solvability of the system. Borrowing the well established techniques of calculating the exact BCS spectrum, we present exact eigenstates and eigenvalues of both the many-body and the effective one-particle Hamiltonians describing the collective neutrino oscillations. For the effective one-body case, we show that spectral splits of neutrinos can be understood in terms of the adiabatic evolution of some quasi-particle degrees of freedom from a high density region where they coincide with flavor eigenstates to the vacuum where they coincide with mass eigenstates. We write down the most general consistency equations which should be satisfied by the effective one-body eigenstates and show that they reduce to the spectral split consistency equations for the appropriate initial conditions.Comment: 26 pages with one figure. Published versio

Dynamical correlation functions of the mesoscopic pairing model

We study the dynamical correlation functions of the Richardson pairing model (also known as the reduced or discrete-state BCS model) in the canonical ensemble. We use the Algebraic Bethe Ansatz formalism, which gives exact expressions for the form factors of the most important observables. By summing these form factors over a relevant set of states, we obtain very precise estimates of the correlation functions, as confirmed by global sum-rules (saturation above 99% in all cases considered). Unlike the case of many other Bethe Ansatz solvable theories, simple two-particle states are sufficient to achieve such saturations, even in the thermodynamic limit. We provide explicit results at half-filling, and discuss their finite-size scaling behavior

On the exactly solvable pairing models for bosons

We propose the new exactly solvable model for bosons corresponding to the attractive pairing interaction. Using the electrostatic analogy, the solution of this model in thermodynamic limit is found. The transition from the superfluid phase with the Bose condensate and the Bogoliubov - type spectrum of excitations in the weak coupling regime to the incompressible phase with the gap in the excitation spectrum in the strong coupling regime is observed.Comment: 19 page

Spin waves in a one-dimensional spinor Bose gas

We study a one-dimensional (iso)spin 1/2 Bose gas with repulsive delta-function interaction by the Bethe Ansatz method and discuss the excitations above the polarized ground state. In addition to phonons the system features spin waves with a quadratic dispersion. We compute analytically and numerically the effective mass of the spin wave and show that the spin transport is greatly suppressed in the strong coupling regime, where the isospin-density (or ``spin-charge'') separation is maximal. Using a hydrodynamic approach, we study spin excitations in a harmonically trapped system and discuss prospects for future studies of two-component ultracold atomic gases.Comment: 4 pages, 1 figur

Dynamical density-density correlations in the one-dimensional Bose gas

The zero-temperature dynamical structure factor of the one-dimensional Bose gas with delta-function interaction (Lieb-Liniger model) is computed using a hybrid theoretical/numerical method based on the exact Bethe Ansatz solution, which allows to interpolate continuously between the weakly-coupled Thomas-Fermi and strongly-coupled Tonks-Girardeau regimes. The results should be experimentally accessible with Bragg spectroscopy.Comment: 4 pages, 3 figures, published versio

New Formula for the Eigenvectors of the Gaudin Model in the sl(3) Case

We propose new formulas for eigenvectors of the Gaudin model in the \sl(3) case. The central point of the construction is the explicit form of some operator P, which is used for derivation of eigenvalues given by the formula $| w_1, w_2) = \sum_{n=0}^\infty P^n/n! | w_1, w_2,0>$, where $w_1$, $w_2$ fulfil the standard well-know Bethe Ansatz equations

Gaudin models solver based on the Bethe ansatz/ordinary differential equations correspondence

We present a numerical approach which allows the solving of Bethe equations whose solutions define the eigenstates of Gaudin models. By focusing on a new set of variables, the canceling divergences which occur for certain values of the coupling strength no longer appear explicitly. The problem is thus reduced to a set of quadratic algebraic equations. The required inverse transformation can then be realized using only linear operations and a standard polynomial root finding algorithm. The method is applied to Richardson's fermionic pairing model, the central spin model and generalized Dicke model.Comment: 10 pages, 3 figures, published versio

Clustering of Bi-Dimensional and Heterogeneous Times Series: Application to Social Sciences Data

We present an application of bi-dimensional and heterogeneous time series clustering in order to resolve a Social Sciences study issue. The dataset is the result of a survey involving more than eight thousand handicapped persons. Sociologists need to know if there are in this dataset some homogeneous classes of people according to two attributes: the idea that handicapped people have about the quality of their life and their couple status (i.e. if they have a partner or not). These two attributes are time series so we had to adapt the k-Means clustering algorithm in order to be efficient with this kind of data. For this purpose, we use the Longest Common Subsequence time series distance for its efficiency to manage time stretching and we extend it to the bidimensional and heterogeneous case. The results of our unsupervised process give some pertinent and surprising clusters that can be easily analyzed by sociologists.Présentation d'une application d'un "bi-dimensional and heterogeneous time series clustering" pour résoudre un problème en sciences sociales. Les données concernent plus de huit mille personnes en situation de handicap. Le problème est de savoir s'il existe de groupes homogènes vis-à-vis de la qualité de vie ressentie et de la vie de couple déclarée. A ces deux séries temporelles, un algorithme de k-Means clustering a été adapté. Nous avons utilisé the Longest Common Subsequence time series distance et nous l'avons étendue au cas bi-dimensionnel et hétérogène. Le résultat a été pertinent et surprenant, utile à l'analyse sociologique

Kernel functions and B\"acklund transformations for relativistic Calogero-Moser and Toda systems

We obtain kernel functions associated with the quantum relativistic Toda systems, both for the periodic version and for the nonperiodic version with its dual. This involves taking limits of previously known results concerning kernel functions for the elliptic and hyperbolic relativistic Calogero-Moser systems. We show that the special kernel functions at issue admit a limit that yields generating functions of B\"acklund transformations for the classical relativistic Calogero-Moser and Toda systems. We also obtain the nonrelativistic counterparts of our results, which tie in with previous results in the literature.Comment: 76 page