On the Lieb-Liniger model in the infinite coupling constant limit

We consider the one-dimensional Lieb-Liniger model (bosons interacting via 2-body delta potentials) in the infinite coupling constant limit (the so-called Tonks-Girardeau model). This model might be relevant as a description of atomic Bose gases confined in a one-dimensional geometry. It is known to have a fermionic spectrum since the N-body wavefunctions have to vanish at coinciding points, and therefore be symmetrizations of fermionic Slater wavefunctions. We argue that in the infinite coupling constant limit the model is indistinguishable from free fermions, i.e., all physically accessible observables are the same as those of free fermions. Therefore, Bose-Einstein condensate experiments at finite energy that preserve the one-dimensional geometry cannot test any bosonic characteristic of such a model

Vortex structures in rotating Bose-Einstein condensates

We present an analytical solution for the vortex lattice in a rapidly rotating trapped Bose-Einstein condensate (BEC) in the lowest Landau level and discuss deviations from the Thomas-Fermi density profile. This solution is exact in the limit of a large number of vortices and is obtained for the cases of circularly symmetric and narrow channel geometries. The latter is realized when the trapping frequencies in the plane perpendicular to the rotation axis are different from each other and the rotation frequency is equal to the smallest of them. This leads to the cancelation of the trapping potential in the direction of the weaker confinement and makes the system infinitely elongated in this direction. For this case we calculate the phase diagram as a function of the interaction strength and rotation frequency and identify the order of quantum phase transitions between the states with a different number of vortex rows.Comment: 17 pages, 12 figures, with addition

Integer Partitions and Exclusion Statistics

We provide a combinatorial description of exclusion statistics in terms of minimal difference $p$ partitions. We compute the probability distribution of the number of parts in a random minimal $p$ partition. It is shown that the bosonic point $p=0$ is a repulsive fixed point for which the limiting distribution has a Gumbel form. For all positive $p$ the distribution is shown to be Gaussian.Comment: 16 pages, 4 .eps figures include

The basic cohomology of the twisted N=16, D=2 super Maxwell theory

We consider a recently proposed two-dimensional Abelian model for a Hodge theory, which is neither a Witten type nor a Schwarz type topological theory. It is argued that this model is not a good candidate for a Hodge theory since, on-shell, the BRST Laplacian vanishes. We show, that this model allows for a natural extension such that the resulting topological theory is of Witten type and can be identified with the twisted N=16, D=2 super Maxwell theory. Furthermore, the underlying basic cohomology preserves the Hodge-type structure and, on-shell, the BRST Laplacian does not vanish.Comment: 9 pages, Latex; new Section 4 showing the invariants added; 2 references and relating remarks adde

Projection on higher Landau levels and non-commutative geometry

The projection of a two dimensional planar system on the higher Landau levels of an external magnetic field is formulated in the language of the non commutative plane and leads to a new class of star products.Comment: 12 pages, latex, corrected versio

Geometric extensions of many-particle Hardy inequalities

Certain many-particle Hardy inequalities are derived in a simple and systematic way using the so-called ground state representation for the Laplacian on a subdomain of $\mathbb{R}^n$. This includes geometric extensions of the standard Hardy inequalities to involve volumes of simplices spanned by a subset of points. Clifford/multilinear algebra is employed to simplify geometric computations. These results and the techniques involved are relevant for classes of exactly solvable quantum systems such as the Calogero-Sutherland models and their higher-dimensional generalizations, as well as for membrane matrix models, and models of more complicated particle interactions of geometric character.Comment: Revised version. 28 page

Universal Hidden Supersymmetry in Classical Mechanics and its Local Extension

We review here a path-integral approach to classical mechanics and explore the geometrical meaning of this construction. In particular we bring to light a universal hidden BRS invariance and its geometrical relevance for the Cartan calculus on symplectic manifolds. Together with this BRS invariance we also show the presence of a universal hidden genuine non-relativistic supersymmetry. In an attempt to understand its geometry we make this susy local following the analogous construction done for the supersymmetric quantum mechanics of Witten.Comment: 6 pages, latex, Volkov Memorial Proceeding

Numerical studies of planar closed random walks

Lattice numerical simulations for planar closed random walks and their winding sectors are presented. The frontiers of the random walks and of their winding sectors have a Hausdorff dimension $d_H=4/3$. However, when properly defined by taking into account the inner 0-winding sectors, the frontiers of the random walks have a Hausdorff dimension $d_H\approx 1.77$.Comment: 15 pages, 15 figure

Finite-size anyons and perturbation theory

We address the problem of finite-size anyons, i.e., composites of charges and finite radius magnetic flux tubes. Making perturbative calculations in this problem meets certain difficulties reminiscent of those in the problem of pointlike anyons. We show how to circumvent these difficulties for anyons of arbitrary spin. The case of spin 1/2 is special because it allows for a direct application of perturbation theory, while for any other spin, a redefinition of the wave function is necessary. We apply the perturbative algorithm to the N-body problem, derive the first-order equation of state and discuss some examples.Comment: 18 pages (RevTex) + 4 PS figures (all included); a new section on equation of state adde