A suggestion for an integrability notion for two dimensional spin systems

TL;DR

This paper proposes a new integrability framework for two-dimensional spin systems based on trialgebraic symmetries, providing explicit conditions and examples linking algebraic structures to physical Hamiltonians.

Contribution

It introduces trialgebraic symmetries as a novel approach to integrability in 2D spin systems and demonstrates their realization in Hamiltonians and Fock spaces.

Findings

Explicit matrix conditions for Hamiltonians with trialgebraic symmetry

Construction of a Hamiltonian realizing a specific trialgebra

Connection between trialgebra symmetries and noncommutative quantum field theory

Abstract

We suggest that trialgebraic symmetries migth be a sensible starting point for a notion of integrability for two dimensional spin systems. For a simple trialgebraic symmetry we give an explicit condition in terms of matrices which a Hamiltonian realizing such a symmetry has to satisfy and give an example of such a Hamiltonian which realizes a trialgebra recently given by the authors in another paper. Besides this, we also show that the same trialgebra can be realized on a kind of Fock space of q-oscillators, i.e. the suggested integrability concept gets via this symmetry a close connection to a kind of noncommutative quantum field theory, paralleling the relation between the integrability of spin chains and two dimensional conformal field theory.