I would like to express the Heisenberg model using the duality analysis  . I show here how to express using Pauli matrices the Ising model https://en.wikipedia.org/wiki/Heisenberg_model_(quantum) but I cannot get the relation $ \sigma _{i}^{z}= \prod_{j\leq i} S_{j}^{x}$ or why $ \sigma _{i}^{x}=S_{i}^{z} S_{i+1}^{z} $  . Also how the $ \sigma _{i}^{y} $ can then be expressed using the duality transition of Pauli matrices?

I would like to express the Heisenberg model using the duality analysis. It is shown here how to express the Ising model using Pauli matrices but I cannot get the relation $ \sigma _{i}^{z}= \prod_{j\leq i} S_{j}^{x}$ or why $ \sigma _{i}^{x}=S_{i}^{z} S_{i+1}^{z} $. Also how can the $ \sigma _{i}^{y} $ then be expressed using the duality transition of Pauli matrices?