Almost local integrable models from supersymmetry algebras

TL;DR

This paper extends supersymmetry algebra methods to construct spectral parameter dependent R-matrices, enabling the creation of integrable models with local or non-local Hamiltonians across various dimensions.

Contribution

It introduces a novel algebraic approach to Baxterize Yang-Baxter solutions from supersymmetry algebras, including almost regular and non-invertible cases, to develop new integrable models.

Findings

Constructed spectral parameter dependent R-matrices from supersymmetry algebra solutions.

Developed integrable models with local and non-local Hamiltonians in multiple dimensions.

Identified new spin systems, including higher spin analogs, from algebraic methods.

Abstract

Supersymmetry algebras can be used to obtain algebraic expressions for constant Yang-Baxter solutions, also known as braid group generators. This was done for non-invertible braid operators in \cite{maity2025non}. In this work we extend this construction for the invertible ones. The resulting expressions are then shown to obey relations analogous to those satisfied by quotients of braid groups. Examples of the latter include the Iwahori-Hecke algebra and the Birman-Murakami-Wenzl (BMW) algebra. As a result, we can Baxterize the constant Yang-Baxter solutions to yield spectral parameter dependent $R$-matrices. The regularity of these $R$-matrices depend on the representation of SUSY generators. In some cases they are regular in the usual sense and in the remaining they are `almost' regular. In the latter case they are also non-invertible. Nevertheless we show that they can still help us construct integrable models in all dimensions of the local Hilbert space. These models can be described by Hamiltonian densities that are either local or non-local, depending on the representation chosen for the SUSY generators. We demonstrate this for all constant $4\times 4$ invertible Yang-Baxter solutions. Apart from finding new nearest-neighbor interaction spin $\frac{1}{2}$ systems, we also find their higher spin analogs due to the algebraic [representation independent] approach.