Calogero-Sutherland Model Research Articles

Exact prefactors in static and dynamic correlation functions of one-dimensional quantum integrable models: Applications to the Calogero-Sutherland, Lieb-Liniger, andXXZmodels

In this article we demonstrate a recently developed technique which addresses the problem of obtaining non-universal prefactors of the correlation functions of 1D systems at zero temperature. Our approach combines the effective field theory description of generic 1D quantum liquids with the finite size scaling of form factors (matrix elements) which are obtained using microscopic techniques developed in the context of integrable models. We thus establish exact analytic forms for the prefactors of the long-distance behavior of equal time correlation functions as well as prefactors of singularities of dynamic response functions. In this article our focus is on three specific integrable models: the Calogero-Sutherland, Lieb-Liniger, and XXZ models.

Conformal blocks in Virasoro and W theories: Duality and the Calogero–Sutherland model

We study the properties of the conformal blocks of the conformal field theories with Virasoro or W-extended symmetry. When the conformal blocks contain only second-order degenerate fields, the conformal blocks obey second order differential equations and they can be interpreted as ground-state wave functions of a trigonometric Calogero–Sutherland Hamiltonian with non-trivial braiding properties. A generalized duality property relates the two types of second order degenerate fields. By studying this duality we found that the excited states of the Calogero–Sutherland Hamiltonian are characterized by two partitions, or in the case of WAk−1 theories by k partitions. By extending the conformal field theories under consideration by a u(1) field, we find that we can put in correspondence the states in the Hilbert state of the extended CFT with the excited non-polynomial eigenstates of the Calogero–Sutherland Hamiltonian. When the action of the Calogero–Sutherland integrals of motion is translated on the Hilbert space, they become identical to the integrals of motion recently discovered by Alba, Fateev, Litvinov and Tarnopolsky in Liouville theory in the context of the AGT conjecture. Upon bosonization, these integrals of motion can be expressed as a sum of two, or in general k, bosonic Calogero–Sutherland Hamiltonian coupled by an interaction term with a triangular structure. For special values of the coupling constant, the conformal blocks can be expressed in terms of Jack polynomials with pairing properties, and they give electron wave functions for special Fractional Quantum Hall states.

Whittaker vectors of the Virasoro algebra in terms of Jack symmetric polynomial

We give an explicit formula of Whittaker vector for Virasoro algebra in terms of the Jack symmetric functions. Our fundamental tools are the Feigin–Fuchs bosonization and the split expression of the Calogero–Sutherland model given by Awata, Matsuo, Odake and Shiraishi.

Source Identity and Kernel Functions for Elliptic Calogero–Sutherland Type Systems

Kernel functions related to quantum many-body systems of Calogero–Sutherland type are discussed, in particular for the elliptic case. The main result is an elliptic generalization of an identity due to Sen that is a source for many such kernel functions. Applications are given, including simple exact eigenfunctions and corresponding eigenvalues of Chalykh–Feigin–Veselov–Sergeev-type deformations of the elliptic Calogero–Sutherland model for special parameter values.

SYMMETRY OF osp(m|n) SPIN CALOGERO–SUTHERLAND MODELS

We introduce osp(m|n) spin Calogero–Sutherland models and find that the models have the symmetry of osp(m|n) half-loop algebra or Yangian of osp(m|n) if and only if the coupling constant of the model equals to .

Shannon and entanglement entropies of one- and two-dimensional critical wave functions

We study the Shannon entropy of the probability distribution resulting from the ground-state wave function of a one-dimensional quantum model. This entropy is related to the entanglement entropy of a Rokhsar-Kivelson-type wave function built from the corresponding two-dimensional classical model. In both critical and massive cases, we observe that it is composed of an extensive part proportional to the length of the system and a subleading universal constant ${S}_{0}$. In $c=1$ critical systems (Tomonaga-Luttinger liquids), we find that ${S}_{0}$ is a simple function of the boson compactification radius. This finding is based on a field-theoretical analysis of the Dyson-Gaudin gas related to dimer and Calogero-Sutherland models. We also performed numerical demonstrations in the dimer models and the spin-1/2 $XXZ$ chain. In a massive (crystal) phase, ${S}_{0}$ is related to the ground-state degeneracy. We also examine this entropy in the Ising chain in a transverse field as an example showing a $c=1/2$ critical point.

Generalized Humbert polynomials and second-order partial differential operators

We define a new class of Humbert's polynomials which generalizes the well-known class of Gegenbauer, Legendre, Pincherle, Horadam–Mahon, Kinney, Hordam–Pethe, Gould and Pathan–Khan polynomials. We prove that these generalized polynomials are eigenfunctions of the Hamiltonian of the Calogero–Sutherland model. We show several differential equations having these polynomials for solutions.

Integrable hydrodynamics of Calogero–Sutherland model: bidirectional Benjamin–Ono equation

We develop a hydrodynamic description of the classical Calogero–Sutherland liquid: a Calogero–Sutherland model with an infinite number of particles and a non-vanishing density of particles. The hydrodynamic equations, being written for the density and velocity fields of the liquid, are shown to be a bidirectional analog of the Benjamin–Ono equation. The latter is known to describe internal waves of deep stratified fluids. We show that the bidirectional Benjamin–Ono equation appears as a real reduction of the modified KP hierarchy. We derive the chiral nonlinear equation which appears as a chiral reduction of the bidirectional equation. The conventional Benjamin–Ono equation is a degeneration of the chiral nonlinear equation at large density. We construct multi-phase solutions of the bidirectional Benjamin–Ono equations and of the chiral nonlinear equations.

Off-diagonal correlations in one-dimensional anyonic models: a replica approach

We propose a generalization of the replica trick that allows us to calculate the largedistance asymptotics of off-diagonal correlation functions in anyonic models with a properfactorizable ground-state wavefunction. We apply this new method to the exactdetermination of all the harmonic terms of the correlations of a gas of impenetrable anyonsand to the Calogero–Sutherland model. Our findings are checked against available analyticand numerical results.

The Hamiltonian of the quantum trigonometric Calogero–Sutherland model in the exceptional algebra E8

We express the Hamiltonian of the quantum trigonometric Calogero–Sutherland model for the Lie algebra E8 and coupling constant κ by using the fundamental irreducible characters of the algebra as dynamical independent variables.

Derivation of Green's function of a spin Calogero–Sutherland model by Uglov's method

The hole propagator of a spin 1/2 Calogero–Sutherland model is derived using Uglov's method, which maps the exact eigenfunctions of the model, called the Yangian Gelfand-Zetlin basis, to a limit of Macdonald polynomials (gl2-Jack polynomials). To apply this mapping method to the calculation of 1-particle Green's function, we confirm that the sum of the field annihilation operator ψ↑ + ψ↓ on a Yangian Gelfand-Zetlin basis is transformed to the field annihilation operator ψ on gl2-Jack polynomials by the mapping. The resultant expression for the hole propagator for a finite-size system is written in terms of renormalized momenta and spin of quasi-holes, and the expression in the thermodynamic limit coincides with the earlier result derived by another method. We also discuss the singularity of the spectral function for a specific coupling parameter where the hole propagator of the spin Calogero–Sutherland model becomes equivalent to the dynamical colour correlation function of an SU(3) Haldane–Shastry model.

Some Orthogonal Polynomials in Four Variables

The symmetric group on 4 letters has the reflection group D3 as an isomor- phic image. This fact follows from the coincidence of the root systems A3 and D3. The isomorphism is used to construct an orthogonal basis of polynomials of 4 variables with 2 parameters. There is an associated quantum Calogero-Sutherland model of 4 identical par- ticles on the line.

The classical hydrodynamics of the Calogero–Sutherland model

We explore the classical version of the mapping, due to Abanov and Wiegmann, of Calogero–Sutherland hydrodynamics onto the Benjamin–Ono equation ‘on the double’. We illustrate the mapping by constructing the soliton solutions to the hydrodynamic equations, and show how certain subtleties arise from the need to include corrections to the naïve replacement of singular sums by principal-part integrals.

Quantum Hamilton–Jacobi Approach to Two Dimensional Singular Oscillator

We obtain the solutions of two-dimensional singular oscillator which is known as the quantum Calogero–Sutherland model both in cartesian and parabolic coordinates within the framework of quantum Hamilton Jacobi formalism. Solvability conditions and eigenfunctions are obtained by using the singularity structures of quantum momentum functions under some conditions. New potentials are generated by using the first two states of singular oscillator for parabolic coordinates.

Explicit computations of low-lying eigenfunctions for the quantum trigonometric Calogero-Sutherland model related to the exceptional algebra E 7

In a previous paper, we studied the characters and Clebsch-Gordan series for the exceptional Lie algebra E7 by relating them to the quantum trigonometric Calogero-Sutherland Hamiltonian with the coupling constant κ = 1. We now extend that approach to the case of an arbitrary coupling constant.

Coherent state of three-particle Calogero-Sutherland model

The three-particle CSM (Calogero-Sutherland model) is solved by the pseudo-angular-momentum method. Analytic expressions of the eigenstate and coherent states are obtained.

Bethe manifolds associated with classical root systems

The Calogero-Sutherland model [1] describes the state of a finite number of nonrelativistic particles on the circle (or on the hyperbola) interacting according to the inverse-square law. For each classical root system (of the type An − 1, Bn, Cn, Dn), we describe the Bethe manifolds related to the Calogero-Sutherland model. In each case, the dimension of the manifold is calculated and its constitutive equations are given.

Entanglement entropy in the Calogero–Sutherland model

We investigate the entanglement entropy between two subsets of particles in the ground state of the Calogero–Sutherland model. By using the duality relations of the Jack symmetric polynomials, we obtain exact expressions for both the reduced density matrix and the entanglement entropy in the limit of an infinite number of particles traced out. From these results, we obtain an upper-bound value of the entanglement entropy. This upper bound has a clear interpretation in terms of fractional exclusion statistics.

Modulation theory of quantum tunneling into a Calogero-Sutherland fluid

Quantum hydrodynamics of interacting electrons with a parabolic single particle spectrum is studied using the Calogero-Sutherland model. The effective action and modulation equations, describing evolution of periodic excitations in the fluid, are derived. Applications to the problem of a single electron tunneling into the FQHE edge state are discussed.

PATH HYPERGEOMETRIC FUNCTIONS

Under a certain condition, we find the explicit formulas for the trace functions of certain intertwining operators among gl(n)-modules, introduced by Etingof in connection with the solutions of the Calogero–Sutherland model. If n = 2, the master function of the trace function is exactly the classical Gauss hypergeometric function. When n > 2, the master functions of the trace functions are a new family of multiple hypergeometric functions, whose differential property and Euler type integral representation are dominated by certain polynomials of integral paths connecting pairs of positive integers. Moreover, we also find the trace functions for sp(2n) explicitly and prove that they give rise to solutions of the Olshanesky–Perelomov model of type C. The master functions of the trace functions for sp(2n) are similar new multiple path hypergeometric functions. Analogous multiple path hypergeometric functions for orthogonal Lie algebras are defined and studied.