426 research outputs found
The Calogero-Sutherland Model and Polynomials with Prescribed Symmetry
The Schr\"odinger operators with exchange terms for certain
Calogero-Sutherland quantum many body systems have eigenfunctions which factor
into the symmetric ground state and a multivariable polynomial. The polynomial
can be chosen to have a prescribed symmetry (i.e. be symmetric or
antisymmetric) with respect to the interchange of some specified variables. For
four particular Calogero-Sutherland systems we construct an eigenoperator for
these polynomials which separates the eigenvalues and establishes
orthogonality. In two of the cases this involves identifying new operators
which commute with the corresponding Schr\"odinger operators. In each case we
express a particular class of the polynomials with prescribed symmetry in a
factored form involving the corresponding symmetric polynomials.Comment: LaTeX 2.09, 31 page
LU factorizations, q=0 limits, and p-adic interpretations of some q-hypergeometric orthogonal polynomials
For little q-Jacobi polynomials and q-Hahn polynomials we give particular
q-hypergeometric series representations in which the termwise q=0 limit can be
taken. When rewritten in matrix form, these series representations can be
viewed as LU factorizations. We develop a general theory of LU factorizations
related to complete systems of orthogonal polynomials with discrete
orthogonality relations which admit a dual system of orthogonal polynomials.
For the q=0 orthogonal limit functions we discuss interpretations on p-adic
spaces. In the little 0-Jacobi case we also discuss product formulas.Comment: changed title, references updated, minor changes matching the version
to appear in Ramanujan J.; 22 p
Various spin-polarization states beyond the maximum-density droplet: a quantum Monte Carlo study
Using variational quantum Monte Carlo method, the effect of Landau-level
mixing on the lowest-energy--state diagram of small quantum dots is studied in
the magnetic field range where the density of magnetic flux quanta just exceeds
the density of electrons. An accurate analytical many-body wave function is
constructed for various angular momentum and spin states in the lowest Landau
level, and Landau-level mixing is then introduced using a Jastrow factor. The
effect of higher Landau levels is shown to be significant; the transition lines
are shifted considerably towards higher values of magnetic field and certain
lowest-energy states vanish altogether.Comment: 4 pages, 2 figures. Submitted to Phys. Rev.
Canted phase in double quantum dots
We perform a Hartree-Fock calculation in order to describe the ground state
of a vertical double quantum dot in the absence of magnetic fields parallel to
the growth direction. Intra- and interdot exchange interactions determine the
singlet or triplet character of the system as the tunneling is tuned. At finite
Zeeman splittings due to in-plane magnetic fields, we observe the continuous
quantum phase transition from ferromagnetic to symmetric phase through a canted
antiferromagnetic state. The latter is obtained even at zero Zeeman energy for
an odd electron number.Comment: 5 pages, 3 figure
Expanded Vandermonde powers and sum rules for the two-dimensional one-component plasma
The two-dimensional one-component plasma (2dOCP) is a system of mobile
particles of the same charge on a surface with a neutralising background.
The Boltzmann factor of the 2dOCP at temperature can be expressed as a
Vandermonde determinant to the power . Recent advances in
the theory of symmetric and anti-symmetric Jack polymonials provide an
efficient way to expand this power of the Vandermonde in their monomial basis,
allowing the computation of several thermodynamic and structural properties of
the 2dOCP for values up to 14 and equal to 4, 6 and 8. In this
work, we explore two applications of this formalism to study the moments of the
pair correlation function of the 2dOCP on a sphere, and the distribution of
radial linear statistics of the 2dOCP in the plane
Kondo Effect of Quantum Dots in the Quantum Hall Regime
Quantum dots in the quantum Hall regime can have pairs of single Slater
determinant states that are degenerate in energy. We argue that these pairs of
many body states may give rise to a Kondo effect which can be mapped into an
ordinary Kondo effect in a fictitious magnetic field. We report on several
properties of this Kondo effect using scaling and numerical renormalization
group analysis. We suggest an experiment to investigate this Kondo effect.Comment: To appear in Phys. Rev. B (5 pages, 4 figures); references added;
several changes in tex
Strongly correlated quantum dots in weak confinement potentials and magnetic fields
We explore a strongly correlated quantum dot in the presence of a weak
confinement potential and a weak magnetic field. Our exact diagonalization
studies show that the groundstate property of such a quantum dot is rather
sensitive to the magnetic field and the strength of the confinement potential.
We have determined rich phase diagrams of these quantum dots. Some experimental
consequences of the obtained phase diagrams are discussed.Comment: 5 pages, 7 figures, new and updated figure
Jack superpolynomials with negative fractional parameter: clustering properties and super-Virasoro ideals
The Jack polynomials P_\lambda^{(\alpha)} at \alpha=-(k+1)/(r-1) indexed by
certain (k,r,N)-admissible partitions are known to span an ideal I^{(k,r)}_N of
the space of symmetric functions in N variables. The ideal I^{(k,r)}_N is
invariant under the action of certain differential operators which include half
the Virasoro algebra. Moreover, the Jack polynomials in I^{(k,r)}_N admit
clusters of size at most k: they vanish when k+1 of their variables are
identified, and they do not vanish when only k of them are identified. We
generalize most of these properties to superspace using orthogonal
eigenfunctions of the supersymmetric extension of the trigonometric
Calogero-Moser-Sutherland model known as Jack superpolynomials. In particular,
we show that the Jack superpolynomials P_{\Lambda}^{(\alpha)} at
\alpha=-(k+1)/(r-1) indexed by certain (k,r,N)-admissible superpartitions span
an ideal {\mathcal I}^{(k,r)}_N of the space of symmetric polynomials in N
commuting variables and N anticommuting variables. We prove that the ideal
{\mathcal I}^{(k,r)}_N is stable with respect to the action of the
negative-half of the super-Virasoro algebra. In addition, we show that the Jack
superpolynomials in {\mathcal I}^{(k,r)}_N vanish when k+1 of their commuting
variables are equal, and conjecture that they do not vanish when only k of them
are identified. This allows us to conclude that the standard Jack polynomials
with prescribed symmetry should satisfy similar clustering properties. Finally,
we conjecture that the elements of {\mathcal I}^{(k,2)}_N provide a basis for
the subspace of symmetric superpolynomials in N variables that vanish when k+1
commuting variables are set equal to each other.Comment: 36 pages; the main changes in v2 are : 1) in the introduction, we
present exceptions to an often made statement concerning the clustering
property of the ordinary Jack polynomials for (k,r,N)-admissible partitions
(see Footnote 2); 2) Conjecture 14 is substantiated with the extensive
computational evidence presented in the new appendix C; 3) the various tests
supporting Conjecture 16 are reporte
Equivalence of the Calogero-Sutherland Model to Free Harmonic Oscillators
A similarity transformation is constructed through which a system of
particles interacting with inverse-square two-body and harmonic potentials in
one dimension, can be mapped identically, to a set of free harmonic
oscillators. This equivalence provides a straightforward method to find the
complete set of eigenfunctions, the exact constants of motion and a linear
algebra associated with this model. It is also demonstrated that
a large class of models with long-range interactions, both in one and higher
dimensions can be made equivalent to decoupled oscillators.Comment: 9 pages, REVTeX, Completely revised, few new equations and references
are adde
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