definitions and properties about the $R$-matrix assuming that the spectral parameter is . We can easily translate them to an spectral parameter:

-   the Yang-Baxter equation reads $$\label{eq:YBE_additive}
      R_{12}\left(z_1-z_2\right) R_{13}\left(z_1-z_3\right) R_{23}\left(z_2-z_3\right) = 
     R_{23}\left(z_2-z_3\right) R_{13}\left(z_1-z_3\right) R_{12}\left(z_1-z_2\right)$$

-   the Markovian property reads ${{\langle\sigma|}} \otimes {{\langle\sigma|}} R(z) = {{\langle\sigma|}} \otimes {{\langle\sigma|}}$

-   the regularity property reads $R(0)=P$

-   the unitarity property reads $R_{12}(z) R_{21}\left(-z\right) = 1$

We are now equipped to construct the inhomogeneous transfer matrix for a model with periodic boundary conditions $$\label{eq:inhomogeneous_transfer_matrix_periodic}
 t(z|\mathbf{z})=tr_0\left( R_{0L}\left(\frac{z}{z_L}\right) \dots R_{02}\left(\frac{z}{z_2}\right)R_{01}\left(\frac{z}{z_1}\right) \right).$$ The transfer matrix acts on the tensor space $\cV^{\otimes L}$ (*i.e* on the whole lattice). The subscripts indicate on which components of the tensor space (*i.e* on which sites of the lattice) the operators are acting non-trivially. Note that an auxiliary space $0$ is introduced and traced out at the end. The main feature of this transfer matrix is that it commutes for different values of the spectral parameter $$\label{eq:transfer_matrix_periodic_commutation}
 [t(x|\mathbf{z}),t(y|\mathbf{z})]=0.$$ It can be shown using the Yang-Baxter equation and the unitarity property .

The transfer matrix is commonly used in the context of exclusion processes (respectively of quantum spin chains) without inhomogeneity parameters, *i.e* with all $z_i=1$. In this setting the first logarithmic 