Sutherland System Research Articles

New Orthogonality Relations for Super-Jack Polynomials and an Associated Lassalle–Nekrasov Correspondence

AbstractThe super-Jack polynomials, introduced by Kerov, Okounkov and Olshanski, are polynomials in$$n+m$$n+mvariables, which reduce to the Jack polynomials when$$n=0$$n=0or$$m=0$$m=0and provide joint eigenfunctions of the quantum integrals of the deformed trigonometric Calogero–Moser–Sutherland system. We prove that the super-Jack polynomials are orthogonal with respect to a bilinear form of the form$$(p,q)\mapsto (L_pq)(0)$$(p,q)↦(Lpq)(0), with$$L_p$$Lpquantum integrals of the deformed rational Calogero–Moser–Sutherland system. In addition, we provide a new proof of the Lassalle–Nekrasov correspondence between deformed trigonometric and rational harmonic Calogero–Moser–Sutherland systems and infer orthogonality of super-Hermite polynomials, which provide joint eigenfunctions of the latter system.

Bispectrality of AG_2 Calogero–Moser–Sutherland System

We consider the generalised Calogero–Moser–Sutherland quantum integrable system associated to the configuration of vectors AG_2, which is a union of the root systems A_2 and G_2. We establish the existence of and construct a suitably defined Baker–Akhiezer function for the system, and we show that it satisfies bispectrality. We also find two corresponding dual difference operators of rational Macdonald–Ruijsenaars type in an explicit form.

Calogero–Sutherland system at a free fermion point

We present two ways to obtain precise expressions for the commuting Hamiltonians of the integrable system regarded as a fermionic limit of the quantum Calogero–Sutherland system as the number of particles tends to infinity with some special values of the coupling constant $$\beta$$ . The construction is realized in the Fock space.

Integrability Aspects of the Current Algebra Representation and the Factorized Quantum Nonlinear Schrödinger Type Dynamical Systems

In this work we study integrability aspects of the current algebra representation and the factorized quantum nonlinear Schrodinger type dynamical systems, initiated before by G. Goldin with collaborators, in suitably renormalized Fock type Hilbert spaces. There is developed the current algebra representation scheme of reconstructing algebraically factorized quantum Hamiltonian and symmetry operators in the Fock type space. There is presented its application to constructing quantum factorized Hamiltonian systems and their symmetry operators in case of quantum integrable spatially many- and one-dimensional dynamical systems. As examples we have studied in detail the factorized structure of Hamiltonian operators, describing such quantum integrable spatially one-dimensional models as the Calogero–Moser–Sutherland and Nonlinear Schrodinger dynamical systems of spin-less Bose-particles.

Noncollinear magnetic structure in U2Pd2In at high magnetic fields

We report an unexpected magnetic-field-driven magnetic structure in the 5f-electron Shastry- Sutherland system U2Pd2In. This phase develops at low temperatures from a noncollinear antiferromagnetic ground state above the critical field of 25.8 T applied along the a-axis. All U moments have a net magnetic moment in the direction of the applied field, described by a ferromagnetic propagation vector qF = (0 0 0) and an antiferromagnetic component described by a propagation vector qAF = (0 0.30 1/2 ) due to a modulation in the direction perpendicular to the applied field. We conclude that this surprising noncollinear magnetic structure is due to a competition between the single-ion anisotropy trying to keep moments, similar to the ground state, along the [110]-type directions, Dzyaloshinskii-Moryia interaction forcing them to be perpendicular to each other and application of the external magnetic field attempting to align them along the field direction.

On the eigenfunctions of hyperbolic quantum Calogero–Moser–Sutherland systems in a Morse potential

We zoom in on special instances of recently found eigenfunctions for the hyperbolic quantum Calogero–Moser–Sutherland n-particle system in a Morse potential. First, dual difference equations in the spectral variable are employed in order to infer that at the discrete spectrum the pertinent eigenfunctions reduce to previously known ones constructed in terms of multivariate Bessel polynomials. Next, it is confirmed that the celebrated multivariate hypergeometric eigenfunctions of the unperturbed hyperbolic Calogero–Moser–Sutherland model are recovered when the Morse potential is switched off.

Bi-Hamiltonian structure of a dynamical system introduced by Braden and Hone

We investigate the finite dimensional dynamical system derived by Braden and Hone in 1996 from the solitons of An−1 affine Toda field theory. This system of evolution equations for an Hermitian matrix L and a real diagonal matrix q with distinct eigenvalues was interpreted as a special case of the spin Ruijsenaars–Schneider models due to Krichever and Zabrodin. A decade later, Li re-derived the model from a general framework built on coboundary dynamical Poisson groupoids. This led to a Hamiltonian description of the gauge invariant content of the model, where the gauge transformations act as conjugations of L by diagonal unitary matrices. Here, we point out that the same dynamics can be interpreted also as a special case of the spin Sutherland systems obtained by reducing the free geodesic motion on symmetric spaces, studied by Pusztai and the author in 2006; the relevant symmetric space being . This construction provides an alternative Hamiltonian interpretation of the Braden–Hone dynamics. We prove that the two Poisson brackets are compatible and yield a bi-Hamiltonian description of the standard commuting flows of the model.

Intertwining operator for AG2 Calogero–Moser–Sutherland system

We consider the generalized Calogero–Moser–Sutherland quantum Hamiltonian H associated with a configuration of vectors AG2 on the plane which is a union of A2 and G2 root systems. The Hamiltonian H depends on one parameter. We find an intertwining operator between H and the Calogero–Moser–Sutherland Hamiltonian for the root system G2. This gives a quantum integral for H of order 6 in an explicit form, thus establishing integrability of H.

Global Description of Action-Angle Duality for a Poisson\u2013Lie Deformation of the Trigonometric varvec{mathrm {BC}_n} Sutherland System

Integrable many-body systems of Ruijsenaars–Schneider–van Diejen type displaying action-angle duality are derived by Hamiltonian reduction of the Heisenberg double of the Poisson–Lie group mathrm{SU}(2n). New global models of the reduced phase space are described, revealing non-trivial features of the two systems in duality with one another. For example, after establishing that the symplectic vector space mathbb {C}^nsimeq mathbb {R}^{2n} underlies both global models, it is seen that for both systems the action variables generate the standard torus action on mathbb {C}^n, and the fixed point of this action corresponds to the unique equilibrium positions of the pertinent systems. The systems in duality are found to be non-degenerate in the sense that the functional dimension of the Poisson algebra of their conserved quantities is equal to half the dimension of the phase space. The dual of the deformed Sutherland system is shown to be a limiting case of a van Diejen system.

Generalized Calogero and Toda Models

A Calogero–Sutherland system with two types of interacting spin variables has been described using the Hitchin approach and quasicompact structure. Complete integrability has been established by means of the Lax equation specified on a singular curve and the classical r-matrix depending on the spectral parameter. Generalized Toda systems have also been considered. Their phase portraits have been described.

Lie Superalgebras and Calogero–Moser–Sutherland Systems

We review recent results obtained at the intersection of the theory of quantum deformed Calogero–Moser–Sutherland systems and the theory of Lie superalgebras. We begin with a definition of admissible deformations of root systems of basic classical Lie superalgebras. For classical series, we prove the existence of Lax pairs. Connections between infinite-dimensional Calogero–Moser–Sutherland systems, deformed quantum CMS systems, and representation theory of Lie superalgebras are discussed.

Calogero–Sutherland system with two types interacting spins

We consider the classical Calogero-Sutherland system with two types of interacting spin variables. It can be reduced to the standard Calogero-Sutherland system, when one of the spin variables vanishes. We describe the model in the Hitchin approach and prove complete integrability of the system by constructing the Lax pair and the classical $r$-matrix with the spectral parameter on a singular curve.

Trigonometric version of quantum–classical duality in integrable systems

We extend the quantum–classical duality to the trigonometric (hyperbolic) case. The duality establishes an explicit relationship between the classical N-body trigonometric Ruijsenaars–Schneider model and the inhomogeneous twisted XXZ spin chain on N sites. Similarly to the rational version, the spin chain data fixes a certain Lagrangian submanifold in the phase space of the classical integrable system. The inhomogeneity parameters are equal to the coordinates of particles while the velocities of classical particles are proportional to the eigenvalues of the spin chain Hamiltonians (residues of the properly normalized transfer matrix). In the rational version of the duality, the action variables of the Ruijsenaars–Schneider model are equal to the twist parameters with some multiplicities defined by quantum (occupation) numbers. In contrast to the rational version, in the trigonometric case there is a splitting of the spectrum of action variables (eigenvalues of the classical Lax matrix). The limit corresponding to the classical Calogero–Sutherland system and quantum trigonometric Gaudin model is also described as well as the XX limit to free fermions.

On a Poisson–Lie deformation of the BCn Sutherland system

A deformation of the classical trigonometric BCn Sutherland system is derived via Hamiltonian reduction of the Heisenberg double of SU(2n). We apply a natural Poisson–Lie analogue of the Kazhdan–Kostant–Sternberg type reduction of the free particle on SU(2n) that leads to the BCn Sutherland system. We prove that this yields a Liouville integrable Hamiltonian system and construct a globally valid model of the smooth reduced phase space wherein the commuting flows are complete. We point out that the reduced system, which contains 3 independent coupling constants besides the deformation parameter, can be recovered (at least on a dense submanifold) as a singular limit of the standard 5-coupling deformation due to van Diejen. Our findings complement and further develop those obtained recently by Marshall on the hyperbolic case by reduction of the Heisenberg double of SU(n,n).

Selberg integrals, super-hypergeometric functions and applications to β-ensembles of random matrices

We study a new Selberg-type integral with n + m indeterminates, which turns out to be related to the deformed Calogero–Sutherland systems. We show that the integral satisfies a holonomic system of n + m non-symmetric linear partial differential equations. We also prove that a particular hypergeometric function defined in terms of super-Jack polynomials is the unique solution of the system. Some properties such as duality relations, integral formulas, Pfaff–Euler and Kummer transformations are also established. As a direct application, we evaluate the expectation value of ratios of characteristic polynomials in the classical β-ensembles of Random Matrix Theory.

Generalized spin Sutherland systems revisited

We present generalizations of the spin Sutherland systems obtained earlier by Blom and Langmann and by Polychronakos in two different ways: from SU(n) Yang–Mills theory on the cylinder and by constraining geodesic motion on the N-fold direct product of SU(n) with itself, for any N>1. Our systems are in correspondence with the Dynkin diagram automorphisms of arbitrary connected and simply connected compact simple Lie groups. We give a finite-dimensional as well as an infinite-dimensional derivation and shed light on the mechanism whereby they lead to the same classical integrable systems. The infinite-dimensional approach, based on twisted current algebras (alias Yang–Mills with twisted boundary conditions), was inspired by the derivation of the spinless Sutherland model due to Gorsky and Nekrasov. The finite-dimensional method relies on Hamiltonian reduction under twisted conjugations of N-fold direct product groups, linking the quantum mechanics of the reduced systems to representation theory similarly as was explored previously in the N=1 case.

On the derivation of Darboux form for the action- angle dual of trigonometric BCn Sutherland system

Recently Fehér and the author have constructed the action-angle dual of the trigonometric BCn Sutherland system via Hamiltonian reduction. In this paper a reduction- based calculation is carried out to verify the canonical Poisson bracket relations on the phase space of this dual model. Hence the material serves complementary purposes whilst it can also be regarded as a suitable modification of the hyperbolic case previously sorted out by Pusztai.

Hidden structures of knot invariants

We discuss a connection of HOMFLY polynomials with Hurwitz covers and represent a generating function for the HOMFLY polynomial of a given knot in all representations as Hurwitz partition function, i.e. the dependence of the HOMFLY polynomials on representation R is naturally captured by symmetric group characters (cut-and-join eigenvalues). The genus expansion and the loop expansion through Vassiliev invariants explicitly demonstrate this phenomenon. We study the genus expansion and discuss its properties. We also consider the loop expansion in details. In particular, we give an algorithm to calculate Vassiliev invariants, give some examples and discuss relations among Vassiliev invariants. Then we consider superpolynomials for torus knots defined via double affine Hecke algebra. We claim that the superpolynomials are not functions of Hurwitz type: symmetric group characters do not provide an adequate linear basis for their expansions. Deformation to superpolynomials is, however, straightforward in the multiplicative basis: the Casimir operators are beta-deformed to Hamiltonians of the Calogero–Moser–Sutherland system. Applying this trick to the genus and Vassiliev expansions, we observe that the deformation is fully straightforward only for the thin knots. Beyond the family of thin knots additional algebraically independent terms appear in the Vassiliev expansions. This can suggest that the superpolynomials do in fact contain more information about knots than the colored HOMFLY and Kauffman polynomials.

On genus expansion of superpolynomials

Recently it was shown that the (Ooguri–Vafa) generating function of HOMFLY polynomials is the Hurwitz partition function, i.e. that the dependence of the HOMFLY polynomials on representation R is naturally captured by symmetric group characters (cut-and-join eigenvalues). The genus expansion and expansion through Vassiliev invariants explicitly demonstrate this phenomenon. In the present paper we claim that the superpolynomials are not functions of such a type: symmetric group characters do not provide an adequate linear basis for their expansions. Deformation to superpolynomials is, however, straightforward in the multiplicative basis: the Casimir operators are β-deformed to Hamiltonians of the Calogero–Moser–Sutherland system. Applying this trick to the genus and Vassiliev expansions, we observe that the deformation is fully straightforward only for the thin knots. Beyond the family of thin knots additional algebraically independent terms appear in the Vassiliev and genus expansions. This can suggest that the superpolynomials do in fact contain more information about knots than the colored HOMFLY and Kauffman polynomials. However, even for the thin knots the beta-deformation is non-innocent: already in the simplest examples it seems inconsistent with the positivity of colored superpolynomials in non-(anti)symmetric representations, which also happens in I. Cherednik's (DAHA-based) approach to the torus knots.

Duality between the trigonometricBCnSutherland system and a completed rational Ruijsenaars–Schneider–van Diejen system

We present a new case of duality between integrable many-body systems, where two systems live on the action-angle phase spaces of each other in such a way that the action variables of each system serve as the particle positions of the other one. Our investigation utilizes an idea that was exploited previously to provide group-theoretic interpretation for several dualities discovered originally by Ruijsenaars. In the group-theoretic framework, one applies Hamiltonian reduction to two Abelian Poisson algebras of invariants on a higher dimensional phase space and identifies their reductions as action and position variables of two integrable systems living on two different models of the single reduced phase space. Taking the cotangent bundle of U(2n) as the upstairs space, we demonstrate how this mechanism leads to a new dual pair involving the BCn trigonometric Sutherland system. Thereby, we generalize earlier results pertaining to the An trigonometric Sutherland system as well as a recent work by Pusztai on the hyperbolic BCn Sutherland system.