Correlation functions of integrable models: a description of the ABACUS algorithm

Recent developments in the theory of integrable models have provided the means of calculating dynamical correlation functions of some important observables in systems such as Heisenberg spin chains and one-dimensional atomic gases. This article explicitly describes how such calculations are generally implemented in the ABACUS C++ library, emphasizing the universality in treatment of different cases coming as a consequence of unifying features within the Bethe Ansatz.Comment: 30 pages, 8 figures, Proceedings of the CRM (Montreal) workshop on Integrable Quantum Systems and Solvable Statistical Mechanics Model

Fermi-Bose transformation for the time-dependent Lieb-Liniger gas

Exact solutions of the Schrodinger equation describing a freely expanding Lieb-Liniger (LL) gas of delta-interacting bosons in one spatial dimension are constructed. The many-body wave function is obtained by transforming a fully antisymmetric (fermionic) time-dependent wave function which obeys the Schrodinger equation for a free gas. This transformation employs a differential Fermi-Bose mapping operator which depends on the strength of the interaction and the number of particles.Comment: 4+ pages, 1 figure; added reference

Geometry of quantum observables and thermodynamics of small systems

The concept of ergodicity---the convergence of the temporal averages of observables to their ensemble averages---is the cornerstone of thermodynamics. The transition from a predictable, integrable behavior to ergodicity is one of the most difficult physical phenomena to treat; the celebrated KAM theorem is the prime example. This Letter is founded on the observation that for many classical and quantum observables, the sum of the ensemble variance of the temporal average and the ensemble average of temporal variance remains constant across the integrability-ergodicity transition. We show that this property induces a particular geometry of quantum observables---Frobenius (also known as Hilbert-Schmidt) one---that naturally encodes all the phenomena associated with the emergence of ergodicity: the Eigenstate Thermalization effect, the decrease in the inverse participation ratio, and the disappearance of the integrals of motion. As an application, we use this geometry to solve a known problem of optimization of the set of conserved quantities---regardless of whether it comes from symmetries or from finite-size effects---to be incorporated in an extended thermodynamical theory of integrable, near-integrable, or mesoscopic systems

Spectra and Symmetry in Nuclear Pairing

We apply the algebraic Bethe ansatz technique to the nuclear pairing problem with orbit dependent coupling constants and degenerate single particle energy levels. We find the exact energies and eigenstates. We show that for a given shell, there are degeneracies between the states corresponding to less and more than half full shell. We also provide a technique to solve the equations of Bethe ansatz.Comment: 15 pages of REVTEX with 2 eps figure

The spin 1/2 Calogero-Gaudin System and its q-Deformation

The spin 1/2 Calogero-Gaudin system and its q-deformation are exactly solved: a complete set of commuting observables is diagonalized, and the corresponding eigenvectors and eigenvalues are explicitly calculated. The method of solution is purely algebraic and relies on the co-algebra simmetry of the model.Comment: 15 page

Noise correlations of the ultra-cold Fermi gas in an optical lattice

In this paper we study the density noise correlations of the two component Fermi gas in optical lattices. Three different type of phases, the BCS-state (Bardeen, Cooper, and Schieffer), the FFLO-state (Fulde, Ferrel, Larkin, and Ovchinnikov), and BP (breach pair) state, are considered. We show how these states differ in their noise correlations. The noise correlations are calculated not only at zero temperature, but also at non-zero temperatures paying particular attention to how much the finite temperature effects might complicate the detection of different phases. Since one-dimensional systems have been shown to be very promising candidates to observe FFLO states, we apply our results also to the computation of correlation signals in a one-dimensional lattice. We find that the density noise correlations reveal important information about the structure of the underlying order parameter as well as about the quasiparticle dispersions.Comment: 25 pages, 11 figures. Some figures are updated and text has been modifie

Discretized vs. continuous models of p-wave interacting fermions in 1D

We present a general mapping between continuous and lattice models of Bose- and Fermi-gases in one dimension, interacting via local two-body interactions. For s-wave interacting bosons we arrive at the Bose-Hubbard model in the weakly interacting, low density regime. The dual problem of p-wave interacting fermions is mapped to the spin-1/2 XXZ model close to the critical point in the highly polarized regime. The mappings are shown to be optimal in the sense that they produce the least error possible for a given discretization length. As an application we examine the ground state of a interacting Fermi gas in a harmonic trap, calculating numerically real-space and momentum-space distributions as well as two-particle correlations. In the analytically known limits the convergence of the results of the lattice model to the continuous one is shown.Comment: 7 pages, 5 figure

Hunting for open clusters in \textit{Gaia} DR2: the Galactic anticentre

The Gaia Data Release 2 (DR2) provided an unprecedented volume of precise astrometric and excellent photometric data. In terms of data mining the Gaia catalogue, machine learning methods have shown to be a powerful tool, for instance in the search for unknown stellar structures. Particularly, supervised and unsupervised learning methods combined together significantly improves the detection rate of open clusters. We systematically scan Gaia DR2 in a region covering the Galactic anticentre and the Perseus arm $(120 \leq l \leq 205$ and $-10 \leq b \leq 10)$, with the goal of finding any open clusters that may exist in this region, and fine tuning a previously proposed methodology successfully applied to TGAS data, adapting it to different density regions. Our methodology uses an unsupervised, density-based, clustering algorithm, DBSCAN, that identifies overdensities in the five-dimensional astrometric parameter space $(l,b,\varpi,\mu_{\alpha^*},\mu_{\delta})$ that may correspond to physical clusters. The overdensities are separated into physical clusters (open clusters) or random statistical clusters using an artificial neural network to recognise the isochrone pattern that open clusters show in a colour magnitude diagram. The method is able to recover more than 75% of the open clusters confirmed in the search area. Moreover, we detected 53 open clusters unknown previous to Gaia DR2, which represents an increase of more than 22% with respect to the already catalogued clusters in this region. We find that the census of nearby open clusters is not complete. Different machine learning methodologies for a blind search of open clusters are complementary to each other; no single method is able to detect 100% of the existing groups. Our methodology has shown to be a reliable tool for the automatic detection of open clusters, designed to be applied to the full Gaia DR2 catalogue.Comment: 8 pages, accepted by Astronomy and Astrophysics (A&A) the 14th May, 2019. Tables 1 and 2 available at the CD

Abundances and kinematics for ten anticentre open clusters

Open clusters are distributed all across the disk and are convenient tracers of its properties. In particular, outer disk clusters bear a key role for the investigation of the chemical evolution of the Galactic disk. The goal of this study is to derive homogeneous elemental abundances for a sample of ten outer disk OCs, and investigate possible links with disk structures such as the Galactic Anticenter Stellar Structure. We analyse high-resolution spectra of red giants, obtained from the HIRES@Keck and UVES@VLT archives. We derive elemental abundances and stellar atmosphere parameters by means of the classical equivalent width method. We also performed orbit integrations using proper motions. The Fe abundances we derive trace a shallow negative radial metallicity gradient of slope -0.027+/-0.007 dex.kpc-1 in the outer 12 kpc of the disk. The [alpha/Fe] gradient appears flat, with a slope of 0.006+/-0.007 dex.kpc-1 . The two outermost clusters (Be 29 and Sau 1) appear to follow elliptical orbits. Be 20 also exhibits a peculiar orbit with a large excursion above the plane. The irregular orbits of the three most metal-poor clusters (of which two are located at the edge of the Galactic disk), if confirmed by more robust astrometric measurements such as those of the Gaia mission, are compatible with an inside-out formation scenario for the Milky Way, in which extragalactic material is accreted onto the outer disk. We cannot determine if Be 20, Be 29,and Sau 1 are of extragalactic origin, as they may be old genuine Galactic clusters whose orbits were perturbed by accretion events or minor mergers in the past 5 Gyr, or they may be representants of the thick disk population. The nature of these objects is intriguing and deserves further investigations in the near future.Comment: 17 pages, 9 figures; accepted for publication in A&