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

TL;DR

This paper introduces a novel algebraic framework linking supersymmetry and quantum integrability in 1D models with 1/r^2 interactions, revealing new hierarchical structures and automatic proofs of integrability.

Contribution

It presents a new algebraic structure that connects supersymmetry with quantum integrability, applicable to exactly solvable 1D models with 1/r^2 interactions, including a hierarchy of models.

Findings

Algebraic structure linking supersymmetry and integrability.

Reduction to ordered Lax equations and harmonic lattice potential.

Hierarchy of models with related super-Hamiltonians.

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.