Factorizing $F$-matrices and the XXZ spin-1/2 chain: A diagrammatic perspective

TL;DR

This paper introduces a diagrammatic approach to factorizing $F$-matrices in the XXZ spin-1/2 chain, providing new visual tools for understanding tensor factorizations and local spin operators.

Contribution

It develops a diagrammatic method to represent and prove the factorization of $F$-matrices and constructs local spin operators in terms of monodromy matrices.

Findings

Diagrams represent $F$-matrix actions in the XXZ chain.

Proof of $F$-matrix factorization of permutation tensors.

Diagrammatic construction of local spin operators.

Abstract

Using notation inherited from the six-vertex model, we construct diagrams that represent the action of the factorizing $F$-matrices associated to the finite length XXZ spin-1/2 chain. We prove that these $F$-matrices factorize the tensor $R^{\sigma}_{1... n}$ corresponding with elements of the permutation group. We consider in particular the diagram for the tensor $R^{\sigma_c}_{1... n}$, which cyclically permutes the spin chain. This leads us to a diagrammatic construction of the local spin operators $S_i^{\pm}$ and $S_i^{z}$ in terms of the monodromy matrix operators.