Super Lax Pairs and Infinite Symmetries in The $1/r^2$ System

Abstract

We present an algebraic structure that provides an interesting and novel link between supersymmetry and quantum integrability. This structure underlies two classes of models that are exactly solvable in 1-dimension and belong to the $1/r^2$ family of interactions. The algebra consists of the commutation between a ``Super- Hamiltonian'', and two other operators, in a Hilbert space that is an enlargement of the original one by introducing fermions. The commutation relations reduce to quantal Ordered Lax equations when projected to the original subspace, and to a statement about the ``Harmonic Lattice Potential'' structure of the Lax operator. These in turn lead to a highly automatic proof of the integrability of these models. In the case of the discrete $SU(n)-1/r^2$ model, the `` Super-Hamiltonian'' is again an $SU(m)-1/r^2$ model with a related $m$, providing an interesting hierarchy of models.

References in corpus (2)

• Lattice Integrable Systems of Haldane-Shastry Type
• Invariants of the Haldane-Shastry $SU(N)$ Chain

Cited by in corpus (50)

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• Hidden Algebras of the (super) Calogero and Sutherland models
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• Calogero-Moser Models V: Supersymmetry and Quantum Lax Pair
• Quantum Calogero-Moser Models: Integrability for all Root Systems
• Supersymmetric Calogero-Moser-Sutherland models and Jack superpolynomials
• Integrals of motion of the Haldane Shastry Model
• Super-Calogero-Moser-Sutherland systems and free super-oscillators : a mapping
• Orthogonality of Jack polynomials in superspace
• Random Matrix Model and the Calogero-Sutherland Model: A Novel Current-Density Mapping
• Superconformal field theory and Jack superpolynomials
• The supersymmetric Ruijsenaars-Schneider model
• Exact Dynamical Correlations of the 1/r^2 Model
• Evaluation and normalization of Jack superpolynomials
• Quantum Lax pairs via Dunkl and Cherednik operators
• Exiton, Spinon and Spin Wave Modes in an Exactly Soluble One-Dimensional Quantum Many-Body System
• Supersymmetric hyperbolic Calogero-Sutherland models by gauging
• Multiparticle SUSY quantum mechanics and the representations of permutation group
• Macdonald polynomials in superspace: conjectural definition and positivity conjectures
• Generalized Hermite polynomials in superspace as eigenfunctions of the supersymmetric rational CMS model
• Super-Calogero model with OSp(2|2) supersymmetry : is the construction unique?
• Intertwining Relations for the Matrix Calogero-like Models: Supersymmetry and Shape Invariance
• Invariants of the $1/r^2$ Supersymmetric t-J Models
• Universal Lax pairs for Spin Calogero-Moser Models and Spin Exchange Models
• Parameter Dependent Commuting Matrices, Plücker relations and Related Quantum Glass Models
• On $\mathcal{N}=2$ supersymmetric Ruijsenaars-Schneider models
• Explicit formulas for the generalized Hermite polynomials in superspace
• The asymptotic Bethe ansatz solution for one-dimensional SU(2) spinor bosons with finite range Gaussian interactions
• Jack superpolynomials with negative fractional parameter: clustering properties and super-Virasoro ideals
• Supersymmetric Many-particle Quantum Systems with Inverse-square Interactions
• $\mathcal{N}=2$ supersymmetric hyperbolic Calogero-Sutherland model
• Dunkl symplectic algebra in generalized Calogero models
• Supersymmetric Embedding of the Quantum Hall Matrix Model
• How coordinate Bethe ansatz works for Inozemtsev model
• Hierarchies of Spin Models related to Calogero-Moser Models
• Exact Heisenberg operator solutions for multi-particle quantum mechanics
• From Jack to Double Jack Polynomials via the Supersymmetric Bridge
• Conservation laws in the continuum $1/r^2$ systems
• Integrable many-body systems of Calogero-Ruijsenaars type
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• General realization of N=4 supersymmetric quantum mechanics and its applications
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• Supersymmetric Calogero-Moser-Sutherland models: superintegrability structure and eigenfunctions
• Equivalence of the super Lax and local Dunkl operators for Calogero-like models
• Automorphisms of the Quantum Cohomology of the Springer Resolution and Applications
• Relativistic generalization of the rational Calogero model